Reilly-type inequality for the Φ-Laplace operator on semislant submanifolds of Sasakian space forms

Abstract

This paper aims to establish new upper bounds for the first positive eigenvalue of the Φ-Laplacian operator on Riemannian manifolds in terms of mean curvature and constant sectional curvature. The first eigenvalue for the Φ-Laplacian operator on closed oriented m-dimensional semislant submanifolds in a Sasakian space form ${\tilde{\mathbb{M}}}^{2k+1}(ϵ)$ is estimated in various ways. Several Reilly-like inequalities are generalized from our findings for Laplacian to the Φ-Laplacian on semislant submanifolds in a sphere ${\mathbb{S}}^{2n+1}$ with $\epsilon =1$ and $\Phi =2$.

1 Introduction and statement of main results

Let $\mathcal{N}^{m}$ be a complete noncompact Riemannian manifold and Σ be the compact domain in $\mathcal{N}^{m}$. Assume $\Lambda _{1}(\Sigma )>0$ denotes the first eigenvalue of the Dirichlet boundary value problem

$$\begin{aligned} \Delta f+\Lambda f=0,\quad \text{in } \Sigma \quad \text{and}\quad f=0 \quad \text{on } \partial \Sigma , \end{aligned}$$

(1.1)

where Δ denotes the Laplace operator on $\mathcal{N}^{m}$. Then, the first eigenvalue $\Lambda _{1}(\mathcal{N})$ is defined by $\Lambda _{1}(\mathcal{N})={inf}_{\Sigma}\Lambda _{1}(\Sigma )$. The Reilly formula relates exclusively to the intrinsic geometry of the manifold and certainly to the specific PDE under consideration. This can be simply understandable with the following example. Let $(\mathcal{N}^{m}, g)$ be a compact m-dimensional Riemannian manifold and let $\Lambda _{1}$ denote the first nonzero eigenvalue of the Neumann problem

$$\begin{aligned} \Delta f+\Lambda _{1} f=0,\quad \text{on } { \mathcal{N}}\quad \text{and} \quad \frac{\partial f}{\partial N}=0,\quad \text{on } \partial{ \mathcal{N}}, \end{aligned}$$

(1.2)

where N is the outward normal on $\partial \mathcal{N}^{m}$. A result of Reilly [22], reads as the following.

Let $\mathcal{N}^{m}$ be a Riemannian manifold and ${\mathbb{R}}^k$ is the Euclidean space having dimensions m and k, respectively. The manifold $\mathcal{N}^{m}$ is connected, closed, and oriented. The $\mathcal{N}^{m}$ is isometrically immersed in ${\mathbb{R}}^k$ with condition $\partial \mathcal{N}^{m}=0$. The mean curvature of this isometric immersion is denoted by $\mathbb{H}$ and the first nonzero eigenvalue $\Lambda _{1}^{\nabla}$ of the Laplacian on $\mathcal{N}^{m}$ can be written as in the sense of Reily [22]

$${\mathrm{\Lambda }}_1^{\mathrm{\nabla }}\le \frac{l}{Vol({\mathcal{N}}^m)}{\int }_{{\mathcal{N}}^m}|\mathbb{H}{|}^{2}dV,$$

(1.3)

where the volume element of $\mathcal{N}^{m}$ is denoted by dV. It can be seen in the literature that many authors were prompted to work in such inequalities for different ambient spaces after the breakthrough of inequality (1.3). In Minkowski spaces, the upper bound for a Finsler submanifold is proposed by both Zeng and He [29]. This upper bound relates to the 1st eigenvalue of the Φ-Laplacian. For a closed manifold, the first eigenvalue of the Φ-Laplace operator is presented by Seto and Wei [25] by using the condition of integral curvature. In the hyperbolic space, the bottom spectra of the Laplace manifold for a complete and a noncompact submanifold is calculated by Lin [19] and the mean curvature has the condition of integral pinching. In addition, Xiong [28] contributed his role on closed hyperspace to find the first Hodge Laplacian eigenvalue. Moreover, Xiong worked for a complete Riemannian manifold that included the Reilly-type sharp upper bounds for the eigenvalues in product manifolds. The generalized Reilly inequality (1.3) and first nonzero eigenvalue of the Φ-Laplace operator is calculated by Du et al. [16]. On a compact submanifold, they used the Wentzel–Laplace operator having a boundary in Euclidean space. Following the same pattern, for Dirichlet and Neumann boundary conditions, Blacker and Seto [6] evidenced a Lichnerowicz-type lower bound for the first nonzero eigenvalue of the Φ-Laplacian. They used the Hessian decomposition on Kaehler manifolds having positive Ricci curvature. A simply connected space form ${\mathbb{M}}^m(c)$ having constant curvature c is obtained by a well-known evaluation for the first nonzero eigenvalue of Laplacian by the immersion of a submanifold $\mathcal{N}^{m}$ in simply connected space having m-dimension. This space form included the Euclidean space ${\mathbb{R}}^m$, the unit sphere ${\mathbb{S}}^m(1)$, and the hyperbolic space $\mathbb{H}{(-1)}^m$ with $c=0, 1$ and $c=-1$, respectively.

In [3, 4, 13, 15], the first nonzero eigenvalue of the Laplacian is evidenced that is considered as the generalization of the results in Reilly [22]. For various ambient spaces, the outcomes of different classes of Riemannian submanifolds indicate that the result of both 1st nonzero eigenvalues depict alike inequalities and ultimately have identical upper bounds [12, 13]. This result is valid for both Dirichlet and Neumann conditions. For an ambient manifold, it is obvious from the literature that Laplace and Φ-Laplace operators on Riemannian submanifolds helped to acquire different breakthroughs in Riemannian geometry (see [5, 8, 9, 11, 14, 17, 20, 21, 23, 26, 29]) through the work of [22]. To define the Φ-Laplacian that is a second-order quasilinear elliptic operator on $\mathcal{N}^{m}$ (compact Riemannian manifold $\mathcal{N}^{m}$ having m-dimension), we have

$$\begin{aligned} \Delta _{\Phi }f=div\bigl( \vert \nabla f \vert ^{\Phi -2}\nabla f\bigr), \end{aligned}$$

(1.4)

where $\Phi >1$ to satisfy the above equation. We have the usual Laplacian for $\Phi =2$. On the other hand, the eigenvalue of $\Delta _{\Phi}$ has similarity with the Laplacian. For instance, if a nonzero function f satisfies the subsequent equation with the Dirichlet boundary condition (1.1) (or Neumann boundary condition (1.2)) then Λ (any real number) is a Dirichlet eigenvalue. Similarly, the above criteria also hold for Neumann boundary conditions (1.2)

$$\begin{aligned} \Delta _{\Phi }f=-\Lambda \vert f \vert ^{\Phi -2}f. \end{aligned}$$

(1.5)

Let us study a Riemannian manifold $\mathcal{N}^{m}$ with no boundary. The Rayleigh-type variational characterization is observed in the first nonzero eigenvalue of $\Delta _{\Phi}$ that is given by $\Lambda _{1, \Phi}$. From (cf. [27]):

$$\begin{aligned} \Lambda _{1,\Phi}=\inf \biggl\{ \frac {\int _{\mathcal{N}} \vert \nabla f \vert ^{q}}{\int _{\mathcal{N}} \Vert f \Vert ^{q}} \Big|f\in W^{1,\Phi}\bigl(\mathcal{N}^{l}\bigr) \{0\}, \int _{\mathcal{N}} \vert f \vert ^{ \Phi -2} f=0 \biggr\} . \end{aligned}$$

(1.6)

This naturally raises the question: Is it possible to generalize the Reilly-type inequalities for submanifolds in spheres through the class of almost contact manifolds that were proved in [1, 13, 15]? In Sasakian space form, our aim is to derive the 1st eigenvalue for the Φ-Laplacian on a slant submanifold. Following this opinion and motivated by the historical development in the analysis of the first nonnull eigenvalue of the Φ-Laplacian on a submanifold in various space forms, by using the Gauss equation and influenced by the studies of [12, 13, 16], our goal is to give a general view of the above Reilly conclusion for the Φ-Laplace operator and we going to provide a sharp estimate of the first eigenvalue for the Φ-Laplacian on a semislant submanifold of Sasakian space form ${\mathbb{M}}^{2k+1}(ϵ)$. The main finding of this paper will be announced in the following theorem.

Theorem 1.1

Let $\mathcal{N}^{m}$ be an $m(\geq 2)$-dimensional closed orientated semislant submanifold in a Sasakian space form ${\tilde{\mathbb{M}}}^{2k+1}(ϵ)$. Then,

• (1) The first nonnull eigenvalue $\Lambda _{1,\Phi}$ of the Φ-Laplacian satisfies:

  $$\begin{array}{rl}{\mathrm{\Lambda }}_{1,\mathrm{\Phi }}& \le \left(\frac{2^{(1-\frac{\mathrm{\Phi }}{2})}{(k+1)}^{(1-\frac{\mathrm{\Phi }}{2})}{m}^{\frac{\mathrm{\Phi }}{2}}}{{(Vol(\mathcal{N}))}^{\mathrm{\Phi }/2}}\right)\\ & \times {\left\{{\int }_{{\mathcal{N}}^m}\{\left(\frac{ϵ+3}{4}\right)+\left(\frac{ϵ-1}{4}\right)\left(\frac{2d_2+d_3(6{cos}^2\vartheta -4)}{m(m-1)}\right)+|\mathbb{H}{|}^2\}dV\right\}}^{\mathrm{\Phi }/2}\\ & \mathit{\text{for }}1<\mathrm{\Phi }\le 2,\end{array}$$

  (1.7)

  and

  $$\begin{array}{rl}{\mathrm{\Lambda }}_{1,\mathrm{\Phi }}& \le \left(\frac{2^{(1-\frac{\mathrm{\Phi }}{2})}{(k+1)}^{(\frac{\mathrm{\Phi }}{2}-1)}{m}^{\frac{\mathrm{\Phi }}{2}}}{Vol(\mathcal{N})}\right)\\ & \times {\int }_{{\mathcal{N}}^m}{\left\{\right|\left(\frac{ϵ+3}{4}\right)+\left(\frac{ϵ-1}{4}\right)\left(\frac{2d_2+d_3(6{cos}^2\vartheta -4)}{m(m-1)}\right)+|\mathbb{H}{|}^2|\}}^{\mathrm{\Phi }/2}dV\\ & \mathit{\text{for }}2<\mathrm{\Phi }\le \frac{m}{2}+1.\end{array}$$

  (1.8)

• (2) The equality carries in (1.7) and (1.8) if and only if $\Phi =2$ and $\mathcal{N}^{m}$ is minimally immersed in a geodesic sphere of radius $r_{\epsilon}$ of ${\tilde{\mathbb{M}}}^{2k+1}(ϵ)$ with the following equalities

  $$\begin{aligned} &r_{0}= \biggl(\frac{m}{\Lambda _{1}^{\Delta}} \biggr)^{1/2}, \\ &r_{1}=\sin ^{-1}{r_{0}}, \\ &r_{-1}=\sinh ^{-1}{r_{0}}. \end{aligned}$$

Remark 1.1

For an immediate consequence of the above, we put $\Phi =2$ in our estimate to find the corollary.

Corollary 1.1

Let $\mathcal{N}^{m}$ be an m-dimensional closed orientated semislant submanifold in Sasakian space form ${\tilde{\mathbb{M}}}^{2k+1}(ϵ)$. Then, $\Lambda _{1}^{\Delta}$ satisfies the following inequality for the Laplacian

$${\mathrm{\Lambda }}_1^{\mathrm{\Delta }}\le \frac{m}{Vol(\mathcal{N})}{\int }_{\mathcal{N}}\{|\mathbb{H}{|}^2+\left(\frac{ϵ+3}{4}\right)+\left(\frac{ϵ-1}{4}\right)\left(\frac{2d_2+d_3(6{cos}^2\vartheta -4)}{m(m-1)}\right)\}dV.$$

(1.9)

The equality’s cases are the same as in Theorem 1.1 (2).

This is an immediate application of Theorem 1.1 by using $1<\Phi \leq 2$, as the Sasakian space form.

Theorem 1.2

Let $\mathcal{N}^{m}$ be an $m(\geq 2)$-dimensional closed orientated semislant submanifold in Sasakian space form ${\tilde{\mathbb{M}}}^{2k+1}(ϵ)$. Then, $\Lambda _{1, \Phi}$ satisfies the following inequality for the Φ-Laplacian

$$\begin{array}{rl}{\mathrm{\Lambda }}_{1,p}& \le \left(\frac{2^{1-\frac{\mathrm{\Phi }}{2}}{(m+1)}^{(1-\frac{\mathrm{\Phi }}{2})}{m}^{\frac{\mathrm{\Phi }}{2}}}{{(Vol(\mathcal{N}))}^{(\mathrm{\Phi }-1)}}\right)\\ & \times {\left\{{\int }_{{\mathcal{N}}^m}{(\left(\frac{ϵ+3}{4}\right)+\left(\frac{ϵ-1}{4}\right)\left(\frac{2d_2+d_3(6{cos}^2\vartheta -4)}{m(m-1)}\right)+|\mathbb{H}{|}^2)}^{\frac{\mathrm{\Phi }}{2(\mathrm{\Phi }-1)}}dV\right\}}^{(\mathrm{\Phi }-1)}\end{array}$$

(1.10)

for $1<\Phi \leq 2$.

Remark 1.2

Consider the inequality (1.10) and put $\Phi =2$, then inequality (1.10) generalizes the Reilly-type inequality (1.9). This shows that the Reilly-type inequality calculates the first eigenvalue for the Laplace operator on a slant submanifold in Euclidean sphere ${\mathbb{S}}^{2k+1}$ (see Theorem 1.2 in [15] and Theorem 1.3 in [13]), are the same in the case of our Theorem 1.1 for $\epsilon =1$ and $\Phi =2$.

2 Preliminaries and notations

An almost contact manifold is an odd-dimensional $C^{\infty}$-manifold $({\tilde{\mathbb{M}}}^{2k+1},g)$ with almost contact structure $(\psi ,\xi ,\eta )$ that satisfies the following properties, i.e.,

$$\begin{aligned} &\psi ^{2}=-I+\eta \otimes \xi, \\ &\eta (\xi )=1, \\ &\psi (\xi )=0, \\ &\eta \circ \psi =0, \end{aligned}$$

(2.1)

$$\begin{aligned} &g(\psi U_{2}, \psi V_{2})=g(U_{2}, V_{2})-\eta (U_{2})\eta (V_{2}), \\ &\eta (U_{2})=g(U_{2}, \xi ) \end{aligned}$$

(2.2)

for any $U_{2}$, $V_{2}$ belong to ${\tilde{\mathbb{M}}}^{2k+1}$. The three parameters of an almost contact structure can be individually elaborated as ψ is a $(1, 1)$-type tensor field, whereas ξ is the structure vector field and η is dual 1-form. In the perspective of the Riemannian connection, an almost contact manifold can be a Sasakian manifold [2, 24] if

$$\begin{aligned} (\widetilde{\nabla}_{U_{2}}\psi )V_{2}=g(U_{2}, V_{2})\xi -\eta (V_{2})U_{2}. \end{aligned}$$

(2.3)

This indicates that

$$\begin{aligned} \widetilde{\nabla}_{U_{2}}\xi =-\psi U_{2}, \end{aligned}$$

(2.4)

where ∇ indicates the Riemannian connection in regard to g and $U_{2}$, $V_{2}$ are any vector fields on ${\tilde{\mathbb{M}}}^{2k+1}$. With this, we consider that ${\tilde{\mathbb{M}}}^{2k+1}$ converts into a Sasakian space form if it has a ψ-sectional constant curvature ϵ and is represented by ${\tilde{\mathbb{M}}}^{2k+1}(ϵ)$. Thus, we can represent the curvature tensor R̃ of ${\tilde{\mathbb{M}}}^{2m+1}(ϵ)$ as:

$$\begin{aligned} \widetilde{R}(X_{2}, Y_{2}, Z_{2}, W_{2})&=\frac{\epsilon +3}{4} \bigl\{ g(Y_{2}, Z_{2})g(X_{2}, W_{2})-g(X_{2}, Z_{2})g(Y_{2}, W_{2}) \bigr\} \\ &\quad {}+\frac{\epsilon -1}{4} \bigl\{ \eta (X_{2})\eta (Z_{2})g(Y_{2}, W_{2})+ \eta (W_{2})\eta (Y_{2})g(X_{2}, Z_{2}) \\ &\quad{}-\eta (Y_{2})\eta (Z_{2})g(X_{2}, W_{2})-\eta (X_{2})g(Y_{2}, Z_{2}) \eta (W_{2}) \\ &\quad{}+g(\psi Y_{2}, Z_{2})g(\psi X_{2}, W_{2})-g(\psi X_{2}, Z_{2})g( \psi Y_{2}, W_{2}) \\ &\quad{}+2g(X_{2}, \psi Y_{2})g(\psi Z_{2}, W_{2}) \bigr\} , \end{aligned}$$

(2.5)

for any arbitary $X_{2}$, $Y_{2}$, $Z_{2}$, $W_{2}$ belonging to ${\tilde{\mathbb{M}}}^{2k+1}$. For more details, see [2, 10, 24].

Assuming that $\mathcal{N}^{m}$ is an m-dimensional submanifold isometrically immersed in a Sasakian space form ${\tilde{\mathbb{M}}}^{2k+1}(ϵ)$, if ∇ and $\nabla ^{\perp}$ are induced connections on the tangent bundle $T\mathcal{N}$ and the normal bundle $T^{\perp}\mathcal{N}$ of $\mathcal{N}$, respectively, then, the Gauss and Weingarten formulas are given by:

$$\begin{aligned} (i) \quad \widetilde{\nabla}_{U_{2}}V_{2}=\nabla _{U_{2}}V_{2}+h(U_{2}, V_{2}),\qquad (ii) \quad \widetilde{\nabla}_{U_{2}}\zeta =-A_{\zeta }{U_{2}}+ \nabla ^{\perp}_{U_{2}} \zeta \end{aligned}$$

(2.6)

for each $U_{2}, V_{2}\in \Gamma (T\mathcal{N})$ and $\zeta \in \Gamma (T^{\perp}\mathcal{N})$, where h and $A_{\zeta}$ are the second fundamental form and shape operator (analogous to the normal vector field ζ), respectively, for the immersion of $\mathcal{N}^{m}$ into ${\tilde{\mathbb{M}}}^{2k+1}(ϵ)$. They are connected as: $g(h(U_{2}, V_{2}), \zeta )=g(A_{\zeta }{U_{2}}, V_{2})$. Throughout the structure vector field ξ is assumed to be tangential to $\mathcal{N}$, otherwise $\mathcal{N}$ is simply antiinvariant. Now, for any $U\in \Gamma (T\mathcal{N})$ and $N\in \Gamma (T^{\perp}\mathcal{N})$, we have:

$$\begin{aligned} (i) \quad \psi {U_{2}}=T{U_{2}}+F{U_{2}}, \qquad (ii) \quad \psi \zeta =t\zeta +f \zeta , \end{aligned}$$

(2.7)

where $T{U_{2}}(t\zeta )$ and $F{U_{2}}(f\zeta )$ are the tangential and normal components of $\psi {U_{2}}(\psi \zeta )$, respectively. From (2.7) it is not difficult to check that for each $U_{2}, V_{2}\in \Gamma (T\mathcal{N})$

$$\begin{aligned} g(T{U_{2}}, V_{2})=-g(U_{2}, TV_{2}). \end{aligned}$$

A submanifold $\mathcal{N}^{m}$ is defined to be a slant submanifold if for any $x\in \mathcal{N}$ and for any vector field $U_{2}\in \Gamma (T\mathcal{N}^{m})$, linearly independent on ξ, the angle between $\psi U_{2}$ and $T\mathcal{N}$ is a constant angle $\vartheta (U_{2})$ that lies between zero and $\pi /2$.

This follows from the definition of slant immersions, where Cabrerizo [7] obtained the necessary and sufficient condition that a submanifold $\mathcal{N}^{m}$ is said to be a slant submanifold if and only if there exists a constant $C\in [0,\pi /2]$ and one tensor fled T is satisfied by the following:

$$\begin{aligned} T^{2}=-C (I-\eta \otimes \xi ), \end{aligned}$$

(2.8)

such that $C=\cos ^{2}\vartheta$. Also, we have a consequence of the above formula

$$\begin{aligned} g(TU_{2}, TV_{2})=\cos ^{2} \vartheta \bigl\{ g(U_{2}, V_{2})-\eta (U_{2}) \eta (V_{2}) \bigr\} . \end{aligned}$$

(2.9)

With the help of the moving-frame method, we explore some of the interesting features of conformal geometry and slant submanifolds. The specific convection has been applied on indices range, though we exclude in a way that:

$$\begin{aligned} 1\leq i,j,s,\ldots \leq m;\qquad m+1\leq \alpha ,\beta ,\gamma , \ldots \leq 2k+1, \qquad 1\leq a, b, c,\ldots \leq 2k+1. \end{aligned}$$

The mean curvature and squared norm of the mean curvature vector $H_{\mathcal{N}}$ of a Riemannian submanifold $\mathcal{N}^{m}$ are defined by:

$$\mathbb{H}=\frac{1}{m}\sum _{i=1}^{n}h(e_i,e_i)\text{and},{\parallel \mathbb{H}\parallel }^2=\frac{1}{m^2}\sum _{r=m+1}^k{\left(\sum _{i=1}^m{h}_{ii}^r\right)}^2.$$

(2.10)

Similarly, the length of the second fundamental form h is given by

$$\begin{aligned} h_{ij}^{r}=g\bigl(h(e_{i}, e_{j}), e_{r}\bigr),\quad \text{and}\quad S=\|h \|^{2}=\sum_{r=m+1}^{k} \sum _{i, j=1}^{n} \bigl(h_{ij}^{r} \bigr)^{2}. \end{aligned}$$

(2.11)

In addition, we denote the following:

$$\begin{aligned} \Vert T \Vert ^{2}=\sum_{i, j=1}^{m}g^{2}(Te_{i}, e_{j}). \end{aligned}$$

(2.12)

Our main motivation comes from the following example:

Example 2.1

([7])

Let $({\mathbb{R}}^{2k+1},\phi ,\xi ,\eta ,g)$ denote the Sasakian manifold with Sasakian structure

$$\begin{aligned} &\eta =\frac{1}{2} \Biggl(dz_{1}-\sum _{i=1}^{k}y^{i}_{1} \,dx^{i}_{1} \Biggr), \qquad \xi =2\frac{\partial}{\partial z_{1}}, \\ &g=\eta \otimes \eta +\frac{1}{4} \Biggl(\sum _{i=1}^{k}\bigl(dx^{i}_{1} \otimes dx^{i}_{1}\otimes dy^{i}_{1} \otimes dy^{i}_{1}\bigr) \Biggr), \\ &\varphi \Biggl(\sum_{i=1}^{k} \biggl(X_{i} \frac{\partial}{\partial x^{i}_{1}}+Y_{i} \frac{\partial}{\partial y^{i}_{1}} \biggr)+Z \frac{\partial}{\partial z_{1}} \Biggr)=\sum_{i=1}^{k} \biggl(Y_{i} \frac{\partial}{\partial y^{i}_{1}}-X_{i} \frac{\partial}{\partial x^{i}_{1}} \biggr)+\sum_{i=1}^{k}Y_{i}y^{i}_{1} \frac{\partial}{\partial z_{1}}, \end{aligned}$$

where $(x^{i}_{1}, y^{i}_{1}, z_{1})$, $i=1\cdots k$ are the coordinates system. It is easy to explain that $({\mathbb{R}}^{2k+1},\phi ,\xi ,\eta ,g)$ is an almost contact metric manifold. Now, consider the 3-dimensional submanifold in ${\mathbb{R}}^5$ with Sasakian structure. For any $\vartheta \in [0, \frac{\pi}{2}]$ such that:

$$\begin{aligned} \psi (u_{1}, v_{1}, t)=2 (u_{1}\cos \vartheta , u_{1}\sin \vartheta , v_{1}, 0, t ). \end{aligned}$$

(2.13)

Under the above immersion $\mathcal{N}^{3}$ is a three-dimensional minimal slant submanifold containing slant angle ϑ and scalar curvature $\tau =-\frac{\cos ^{2}\vartheta}{3}$.

Similarly, we give more examples for a nonminimal submanifold.

Example 2.2

([7])

For any constant λ, we define an immersion:

$$\begin{aligned} \psi (u_{1}, v_{1}, t)=2 \bigl(e^{\lambda u_{1}} \cos u_{1}\cos v_{1}, e^{ \lambda u_{1}}\sin u_{1}\cos v_{1}, e^{\lambda u_{1}}\cos u_{1}\sin v_{1}, e^{\lambda u_{1}}\sin u_{1}\sin v_{1}, t \bigr). \end{aligned}$$

(2.14)

It is easy to see that the above immersion is a three-dimensional slant submanifold with slant angle $\vartheta =\cos ^{-1} (\frac{|\lambda |}{\sqrt{1+\lambda ^{2}}} )$. Moreover, scalar curvature $\tau =-\frac{\lambda ^{2}}{3(1+\lambda ^{2})}$ and mean curvature $|\mathbb{H}|=\frac{2e^{-\lambda u_1}}{3\sqrt{1+{\lambda }^2}}$.

A Riemannian submanifold $\mathcal{N}^{m}$ of an almost contact manifold $\tilde{\mathbb{M}}$ is said to be a semislant submanifold if there exist two orthogonal distributions $\mathcal{D}$ and $\mathcal{D}^{\vartheta}$ such that $T\mathcal{N}=\mathcal{D}\oplus \mathcal{D}^{\vartheta} \oplus \zeta$, the distribution $\mathcal{D}$ is invariant, i.e., $\varphi \mathcal{D}=\mathcal{D}$ and the distribution $D^{\theta}$ is slanted with slant angle $\vartheta \neq \frac{\pi}{2}$. If we denote the dimensions of $\mathcal{D}$ and $\mathcal{D}^{\theta}$ by $d_{2}$ and $d_{3}$, respectively, then it is clear that contact CR-submanifolds and slant submanifolds are semislant submanifolds with $\theta = \frac{\pi}{2}$ and $d_{2} = 0$, respectively. If neither $d_{2} = 0$ nor $\theta = \frac{\pi}{2}$, then $\mathcal{N}^{m}$ is a proper semislant submanifold.

Remark 2.1

It is clear that a semislant submanifold is generalized to a slant submanifold with $d_{2}=0$.

Remark 2.2

A totally real submanifold is a particular case of a semislant submanifold with slant angle $\vartheta =\frac{\pi}{2}$ and $d_{2}=0$.

It is necessary to clarify the definition of the curvature tensor R̃ for a slant submanifold in the Sasakian space form ${\tilde{\mathbb{M}}}^{2k+1}(ϵ)$ and is given by:

$$\begin{aligned} \widetilde{R}(e_{i}, e_{j}, e_{i}, e_{j})&= \biggl(\frac{\epsilon +3}{4} \biggr) \bigl(m^{2}-m \bigr) \\ &\quad{}+ \biggl(\frac{\epsilon -1}{4} \biggr) \Biggl\{ 3\sum _{ii, j=1}^{m}g^{2}( \varphi e_{i}, e_{j})-2(m-1) \Biggr\} . \end{aligned}$$

(2.15)

On the other hand, let $\{e_{1},\ldots e_{d_{2}},\ldots e_{m}=\zeta \}$ be an orthonormal basis of $T_{x}\mathcal{N}$ such that

$$\begin{aligned} &e_{1},e_{2}=\varphi e_{1},\ldots ,e_{2d_{2}-1},e_{2d_{2}}=\varphi e_{2d_{2}-1}, \\ &e_{2d_{2}+1},e_{2d_{2}+2}=\sec \vartheta Te_{2d_{2}+1},\ldots ,e_{2d_{2}+2d_{3}-1},e_{2d_{2}+2d_{3}}= \sec \vartheta Te_{2d_{2}+2d_{3}-1},\ldots , \\ &\vdots \\ &e_{2d_{2}+2d_{3}},e_{2d_{2}+2d_{2}+1}=\zeta . \end{aligned}$$

Thus, we have

$$\begin{aligned} g(\varphi e_{1}, e_{2})=g(\varphi e_{1}, \sec \vartheta Te_{1})=\sec \vartheta g(\varphi e_{1}, Te_{1})=\sec \vartheta g(T e_{1}, Te_{1}). \end{aligned}$$

(2.16)

It is clear that the dimension of $\mathcal{N}^{m}$ can be decomposed as $m=2d_{2}+2d_{3}+1$. Then, from (2.9), we derive that:

$$\begin{aligned} g(\varphi e_{1}, e_{2})=\cos \vartheta . \end{aligned}$$

(2.17)

In similar way, we repeat that then:

$$\begin{aligned} g^{2}(\varphi e_{i}, e_{i+1})=\cos ^{2}\vartheta \quad \implies\quad \sum_{i, j=1}^{m}g^{2}( \varphi e_{i}, e_{j})=2 \bigl(d_{2}+d_{3} \cos ^{2}\vartheta \bigr). \end{aligned}$$

(2.18)

Merging (2.15) and (2.18) implies that

$$\begin{aligned} \widetilde{R}(e_{i}, e_{j}, e_{i}, e_{j})&= \biggl(\frac{\epsilon +3}{4} \biggr) \bigl(m^{2}-m \bigr) \\ &\quad{}+ \biggl(\frac{\epsilon -1}{4} \biggr) \bigl\{ 2d_{2}+d_{3} \bigl(6\cos ^{2} \vartheta -4 \bigr) \bigr\} . \end{aligned}$$

(2.19)

2.1 Structure equations for semislant submanifolds

Let x be a totally real embedding from $\mathcal{N}^{m}$ to an $2k+1$-dimensional Riemannian manifold $(\tilde{\mathbb{M}},\tilde{g})$. Then, $\mathcal{N}^{m}$ has an induced metric $g_{\mathcal{N}}=x^{*}\widetilde{g}$. Let us consider ${\tilde{\mathbb{M}}}^{2k+1}={\tilde{\mathbb{M}}}^{2k+1}(ϵ)$, then pulling back [[1] Eq. (12)] by x and using [[1] Eqs. (13), (14)], we obtain the Gauss equations for a slant submanifold in Sasakian space form ${\tilde{\mathbb{M}}}^{2k+1}(ϵ)$ and taking into account (2.15)

$$\begin{aligned} R_{ijtl}&= \biggl(\frac{\epsilon +3}{4} \biggr) (\delta _{it}\delta _{jl}- \delta _{il}\delta _{jt})+ \biggl(\frac{\epsilon -1}{4} \biggr) \bigl\{ 3( \varphi e_{i}, e_{j})-2(m-1) \bigr\} \\ &\quad{}+\sum_{\alpha } \bigl(h_{it}^{\alpha }h_{jl}^{\alpha}-h_{il}^{ \alpha }h_{jt}^{\alpha} \bigr). \end{aligned}$$

(2.20)

Taking the trace of the above equation and using (2.19), we obtain:

$$R=m^2|\mathbb{H}{|}^2-S+\left(\frac{ϵ+3}{4}\right)m(m-1)+\left(\frac{ϵ-1}{4}\right)\{2d_2+d_3(6{cos}^2\vartheta -4)\},$$

(2.21)

where R is the scalar curvature of $\mathcal{N}^{m}$ and S is the length of the second fundamental form h.

2.2 Conformal relations

In this section, we will look at how the conformal transformation affects the curvature and the second fundamental form. Although these relationships are well known (cf. [1]), we use the moving-frame method to provide a quick proof for the readers’ convenience.

Assume that ${\tilde{\mathbb{M}}}^{2k+1}$ has a new metric $\tilde{{g}}=e^{2\rho}\tilde{g}$, that is conformal to g̃, and where $\rho \in C^{\mathrm{\infty }}(\tilde{\mathbb{M}})$. Then, $\tilde{\varpi}_{a}=e\rho \varpi _{a}$ is the dual coframe of $(\tilde{\mathbb{M}},\tilde{g})$, and $\tilde{e}_{a}=e^{\rho }e_{a}$ is the orthogonal frame of $(\tilde{\mathbb{M}},\tilde{g})$. The equality’s equations of $(\tilde{\mathbb{M}},\tilde{g})$ are given in ([1], Eqs. (20), (21), (22) (23)) by:

$$\begin{aligned} \widetilde{\varpi}_{ab}=\varpi _{ab}+\rho _{a}\varpi _{b} -\rho _{b} \varpi _{a}, \end{aligned}$$

(2.22)

where $\rho _{a}$ is the covariant derivative of ρ with respect to $e_{a}$, that is, $d\rho =\sum_{a}\rho _{a} e_{a}$.

$$\begin{aligned} e^{2\rho}\widetilde{R}_{ijtl}&= R_{ijtl}- (\rho _{it}\delta _{jl}+ \rho _{jl}\delta _{it}-\rho _{il}\delta _{jt}-\rho _{jt}\delta _{il} ) \\ &\quad{}+ (\rho _{i}\rho _{t}\delta _{jl}+\rho _{j}\rho _{l}\delta _{it}- \rho _{j} \rho _{t}\delta _{il}-\rho _{i}\rho _{l}\delta _{jt} ) \\ &\quad{}- \vert \nabla _{\alpha} \vert ^{2} (\delta _{it}\delta _{jl}-\delta _{il} \delta _{jt} ). \end{aligned}$$

(2.23)

By pulling back (2.22) to $\mathcal{N}^{m}$ by x, we have:

$$\begin{array}{rl}& {\tilde{h}}_{ij}^{\alpha }=e^{-\rho }(h_{ij}^{\alpha }-{\rho }_{\alpha }{\delta }_{ji}),\\ & {\tilde{\mathbb{H}}}^{\alpha }=e^{-\alpha }({\mathbb{H}}^{\alpha }-{\rho }_{\alpha }),\end{array}$$

(2.24)

from this, it is easy to obtain the useful relation:

$$e^{2\rho }(\tilde{S}-m|\tilde{\mathbb{H}}{|}^2)+m|\mathbb{H}{|}^2=S.$$

(2.25)

3 Proof of main result

In this section we shall prove Theorem 1.1 announced in a previous section. First, some fundamental formulas will be presented and some useful lemmas from [20] will be recalled to our setting. For the proposes of this paper, we are going to provide an important lemma that was essentially motivated by the study in [1, 20].

Remark 3.1

A simply connected Sasakian space form ${\mathbb{M}}^{2k+1}$ is a $(2k+1)$-sphere ${\mathbb{S}}^{2k+1}$ and Euclidean space ${\mathbb{R}}^{2k+1}$ with constant φ-sectional curvature $\epsilon =1$ and $\epsilon =-3$, respectively.

Based on the above arguments, we have a lemma.

Lemma 3.1

([1])

Let $\mathcal{N}^{m}$ be a slant submanifold of Sasakian space form ${\tilde{\mathbb{M}}}^{2k+1}(ϵ)$ that is closed and oriented with dimension m. If $x:{\mathcal{N}}^m\to {\tilde{\mathbb{M}}}^{2k+1}(ϵ)$ is an embedding from $\mathcal{N}^{m}$ into ${\tilde{\mathbb{M}}}^{2k+1}(ϵ)$, then there exists a regular conformal map $\mathrm{\Gamma }:{\tilde{\mathbb{M}}}^{2k+1}(ϵ)\to {\mathbb{S}}^{2k+1}(1)\subset {\mathbb{R}}^{2k+2}$ such that the embedding $\varpi =\Gamma \circ x=(\varpi ^{1},\ldots \varpi ^{2k+2})$ satisfies that:

$$\begin{aligned} \int _{\mathcal{N}^{m}} \bigl\vert \varpi ^{a} \bigr\vert ^{\Phi -2}\varpi ^{a} \,dV_{ \mathcal{N}}=0,\quad a=1,\ldots 2(k+1), \end{aligned}$$

(3.1)

for $\Phi >1$.

In the above Lemma 3.1 by the constructed test function, we produce an upper bound for $\Lambda _{1,\Phi}$ in the form of the conformal function that is comparable with Lemma 2.7 in [20].

Proposition 3.1

Let $\mathcal{N}^{n}$ be an $m\geq 2$-dimensional closed orientated slant submanifold into Sasakian space form ${\tilde{\mathbb{M}}}^{2k+1}(ϵ)$. Then we have,

$$\begin{aligned} \Lambda _{1, \Phi}Vol\bigl(\mathcal{N}^{m}\bigr) \leq 2^{|1-\frac{\Phi}{2}|}(k+1)^{|1- \frac{\Phi}{2}|}m^{\frac{\Phi}{2}} \int _{\mathcal{N}^{m}}\bigl(e^{2\rho} \bigr)^{ \frac{\Phi}{2}}\,dV, \end{aligned}$$

(3.2)

where Γ is the conformal map in Lemma 3.1and for all $\Phi >1$. Identified by $\Upsilon _{\varepsilon}$ is the standard metric on ${\tilde{\mathbb{M}}}^{2k+1}(ϵ)$ and it is considered that $\Gamma ^{*}\Upsilon _{1}=e^{2\rho}\Upsilon _{\varepsilon}$,

Proof

Considering $\varpi ^{a}$ as a test function along with Lemma 3.1, we derive

$$\begin{aligned} \Lambda _{1,\Phi} \int _{\mathcal{N}^{m}} \bigl\vert \varpi ^{a} \bigr\vert ^{\Phi}\leq \bigl\vert \nabla \varpi ^{a} \bigr\vert ^{\Phi}\,dV,\quad 1\leq a\leq 2(k+1). \end{aligned}$$

(3.3)

Observe that $\sum_{a=1}^{2k+2}|\varpi ^{a}|^{2}=1$, then $|\varpi ^{a}|\leq 1$. We accomplish:

$$\begin{aligned} \sum_{a=1}^{2k+2} \bigl\vert \nabla \varpi ^{a} \bigr\vert ^{2}=\sum _{i=1}^{m} \vert \nabla _{e_{i}}\varpi \vert ^{2}=me^{2\rho}. \end{aligned}$$

(3.4)

By using $1<\Phi \leq 2$, then we derive:

$$\begin{aligned} \bigl\vert \varpi ^{a} \bigr\vert ^{2} \leq \bigl\vert \varpi ^{a} \bigr\vert ^{\Phi}. \end{aligned}$$

(3.5)

Using the Holder inequality along with (3.3), (3.4), and (3.5), we are able to obtain

$$\begin{aligned} \Lambda _{1,\Phi} Vol(\mathcal{N})&=\Lambda _{1,\Phi}\sum _{a=1}^{2k+2} \int _{\mathcal{N}^{m}} \bigl\vert \varpi ^{a} \bigr\vert ^{2}\,dV\leq \Lambda _{1,\Phi}\sum_{a=1}^{2k+2} \int _{\mathcal{N}^{m}} \bigl\vert \varpi ^{a} \bigr\vert ^{\Phi }\,dV \\ &\leq \Lambda _{1,\Phi} \int _{\mathcal{N}^{m}}\sum_{a=1}^{2k+2} \bigl\vert \nabla \varpi ^{a} \bigr\vert ^{\Phi }\,dV\leq (2k+2)^{1-\Phi /2} \int _{ \mathcal{N}^{m}} \Biggl(\sum_{a=1}^{m} \bigl\vert \nabla \varpi ^{a} \bigr\vert ^{2} \Biggr)^{\frac{\Phi}{2}}\,dV \\ &=2^{1-\frac{\Phi}{2}}(k+1)^{1-\frac{\Phi}{2}} \int _{\mathcal{N}^{m}} \bigl(me^{2\rho} \bigr)^{\frac{\Phi}{2}}\,dV. \end{aligned}$$

This gives us the desired outcome (3.2). On the contrary, if we assume $\Phi \geq 2$, then by applying the Holder inequality we have

$$\begin{aligned} 1=\sum_{a=1}^{2k+2} \bigl\vert \varpi ^{a} \bigr\vert ^{2}\leq (2k+2)^{1- \frac{2}{\Phi}} \Biggl(\sum_{a=1}^{2k+2} \bigl\vert \varpi ^{a} \bigr\vert ^{\Phi} \Biggr)^{\frac{2}{\Phi}}. \end{aligned}$$

(3.6)

The outcome we obtain is

$$\begin{aligned} \Lambda _{1,\Phi}Vol\bigl(\mathcal{N}^{m}\bigr) \leq (2k+2)^{\frac{\Phi}{2}-1} \Biggl(\sum_{a=1}^{2k+2} \Lambda _{1,\Phi} \int _{\mathcal{N}^{m}} \bigl\vert \varpi ^{a} \bigr\vert ^{\Phi }\,dV \Biggr). \end{aligned}$$

(3.7)

The Minkowski inequality gives

$$\begin{aligned} \sum_{a=1}^{2k+2} \bigl\vert \nabla \varpi ^{a} \bigr\vert ^{\Phi}\leq \Biggl( \sum_{a=1}^{2k+2} \bigl\vert \nabla \varpi ^{a} \bigr\vert ^{2} \Biggr)^{ \frac{\Phi}{2}}= \bigl(me^{2\rho} \bigr)^{\frac{\Phi}{2}}. \end{aligned}$$

(3.8)

Hence, (3.2) follows from (3.3), (3.7), and (3.8). This completes the proof of the proposition. □

We are now in a position to prove Theorem 1.1.

3.1 Proof of Theorem 1.1

To begin with $1<\Phi \leq 2$, then $\frac{\Phi}{2}\leq 1$. Taking help from Proposition 3.1 and implementing the Hölder inequality, we have:

$$\begin{aligned} \Lambda _{1, \Phi} Vol\bigl(\mathcal{N}^{m}\bigr)& \leq2^{1-\frac{\Phi}{2}}(k+1)^{1- \frac{\Phi}{2}}m^{\frac{\Phi}{2}} \int _{\mathcal{N}^{m}} \bigl(e^{2 \rho} \bigr)^{\frac{\Phi}{2}}\,dV \\ &\leq 2^{1-\frac{\Phi}{2}}(k+1)^{1-\frac{\Phi}{2}|}m^{\frac{\Phi}{2}}\bigl(Vol( \mathcal{N})\bigr)^{1-\frac{\Phi}{2}} \biggl( \int _{\mathcal{N}^{m}} e^{2 \rho}\,dV \biggr)^{\frac{\Phi}{2}}. \end{aligned}$$

By using both conformal relations and Gauss equations, it is possible to calculate $e^{2\rho}$. Let ${\tilde{\mathbb{M}}}^{2k+1}={\tilde{\mathbb{M}}}^{2k+1}(ϵ)$, and $\tilde{g}=e^{-2\rho} \Upsilon _{ \epsilon}$, $\tilde{{g}}=\Gamma ^{*}\Upsilon _{1}$ in the above. From (2.21), the Gauss equations for the embedding x and the slant embedding $\varpi =\Gamma \circ x$ are, respectively:

$$\begin{array}{rl}R& =\left(\frac{ϵ+3}{4}\right)m(m-1)+\left(\frac{ϵ-1}{4}\right)\{2d_2+d_3(6{cos}^2\vartheta -4)\}\\ & +m(m-1)|\mathbb{H}{|}^2+(m|\mathbb{H}{|}^2-S),\end{array}$$

(3.9)

$$\tilde{R}=m(m-1)+m(m-1)|\tilde{\mathbb{H}}{|}^2+(m|\tilde{\mathbb{H}}{|}^2-\tilde{S}).$$

(3.10)

Tracing (2.23), it can be established that:

$$\begin{aligned} e^{2\rho}\tilde{R}= R-(m-2) (m-1) \vert \nabla _{\rho} \vert ^{2}-2(m-1)\Delta _{ \rho}, \end{aligned}$$

(3.11)

which together with replacement of (3.9) and (3.10) into (3.11) gives:

$$\begin{array}{rl}& e^{2\rho }(m(m-1)+m(m-1)|\tilde{\mathbb{H}}{|}^2+(m|\tilde{\mathbb{H}}{|}^2-\tilde{S}))\\ & =\left(\frac{ϵ+3}{4}\right)m(m-1)\\ & +\left(\frac{ϵ-1}{4}\right)\{2d_2+d_3(6{cos}^2\vartheta -4)\}\\ & +m(m-1)|\mathbb{H}{|}^2+(m|\mathbb{H}{|}^2-S)\\ & -(m-2)(m-1)|{\mathrm{\nabla }}_{\rho }{|}^2-2(m-1){\mathrm{\Delta }}_{\rho }.\end{array}$$

This implies the following:

$$\begin{array}{rl}& e^{2\rho }\tilde{S}-S-(m-2)(m-1)|{\mathrm{\nabla }}_{\rho }{|}^2-2(m-1){\mathrm{\Delta }}_{\rho }\\ & =m(m-1)\{\{e^{2\rho }-\left(\frac{ϵ+3}{4}\right)-\left(\frac{ϵ-1}{4}\right)\left(\frac{2d_2+d_3(6{cos}^2\vartheta -4)}{m(m-1)}\right)\}\\ & +(e^{2\rho }|\tilde{\mathbb{H}}{|}^2-|\mathbb{H}{|}^2)\}+m(e^{2\rho }|\tilde{\mathbb{H}}{|}^2-|\mathbb{H}{|}^2).\end{array}$$

Now, from (2.24) and (2.25), we derive:

$$\begin{array}{rl}& m(m-1)\{e^{2\rho }-\left(\frac{ϵ+3}{4}\right)-\left(\frac{ϵ-1}{4}\right)\left(\frac{2d_2+d_3(6{cos}^2\vartheta -4)}{m(m-1)}\right)\}\\ & +m(m-1)\sum _{\alpha }{({\mathbb{H}}^{\alpha }-{\rho }_{\alpha })}^2\\ & =m(m-1)|\mathbb{H}{|}^2-(m-2)(m-1)|{\mathrm{\nabla }}_{\rho }{|}^2-2(m-1){\mathrm{\Delta }}_{\rho }.\end{array}$$

Dividing by $m(m-1)$ in the above equation, it implies that

$$\begin{array}{rl}{e}^{2\rho }& =\{\left(\frac{ϵ+3}{4}\right)+\left(\frac{ϵ-1}{4}\right)\left(\frac{2d_2+d_3(6{cos}^2\vartheta -4)}{m(m-1)}\right)+|\mathbb{H}{|}^2\}\\ & -\frac{2}{m}{\mathrm{\Delta }}_{\rho }-\frac{m-2}{m}|{\mathrm{\Delta }}_{\rho }{|}^2-|{({\tilde{\mathrm{\nabla }}}_{\rho })}^{\mathrm{\perp }}-\mathbb{H}{|}^2.\end{array}$$

(3.12)

Taking integration along dV, it is not complicated to obtain the following

$$\begin{array}{rl}& {\mathrm{\Lambda }}_{1,\mathrm{\Phi }}Vol\left({\mathcal{N}}^m\right)\\ & \le 2^{1-\frac{\mathrm{\Phi }}{2}}{(k+1)}^{1-\frac{\mathrm{\Phi }}{2}|}{m}^{\frac{\mathrm{\Phi }}{2}}{\left(Vol\left({\mathcal{N}}^m\right)\right)}^{1-\frac{\mathrm{\Phi }}{2}}{\left({\int }_{{\mathcal{N}}^m}{e}^{2\rho }dV\right)}^{\frac{\mathrm{\Phi }}{2}}\\ & \le \frac{{(2k+2)}^{1-\frac{\mathrm{\Phi }}{2}|}{m}^{\frac{\mathrm{\Phi }}{2}}}{{(Vol(\mathcal{N}))}^{\frac{\mathrm{\Phi }}{2}-1}}\\ & \times {\left\{{\int }_{{\mathcal{N}}^m}\{\left(\frac{ϵ+3}{4}\right)+\left(\frac{ϵ-1}{4}\right)\left(\frac{2d_2+d_3(6{cos}^2\vartheta -4)}{m(m-1)}\right)+|\mathbb{H}{|}^2\}dV\right\}}^{\frac{\mathrm{\Phi }}{2}}.\end{array}$$

The above result is comparable to (1.7) as we desired to prove. In the case where $\Phi >2$, it is not possible to apply the Holder inequality directly to govern $\int _{\mathcal{N}}(e^{2\nu})^{\frac{\Phi}{2}}$ by using $\int _{\mathcal{N}}(e^{2\rho})$. We did multiply both sides of (3.12) with the factor $e^{(\Phi -2)\rho}$ and then solve by using integration on $\mathcal{N}^{m}$ (cf. [11])

$$\begin{array}{rl}{\int }_{{\mathcal{N}}^m}{e}^{\mathrm{\Phi }\rho }dV& \le {\int }_{{\mathcal{N}}^m}\{\left(\frac{ϵ+3}{4}\right)+\left(\frac{ϵ-1}{4}\right)\left(\frac{2d_2+d_3(6{cos}^2\vartheta -4)}{m(m-1)}\right)+|\mathbb{H}{|}^2\}{e}^{(\mathrm{\Phi }-2)\rho }dV\\ & -\left(\frac{m-2-2\mathrm{\Phi }+4}{m}\right){\int }_{\mathcal{N}}{e}^{(\mathrm{\Phi }-2)}|{\mathrm{\Delta }}_{\rho }{|}^{2}dV\\ & \le {\int }_{{\mathcal{N}}^m}\{\left(\frac{ϵ+3}{4}\right)+\left(\frac{ϵ-1}{4}\right)\left(\frac{2d_2+d_3(6{cos}^2\vartheta -4)}{m(m-1)}\right)+|\mathbb{H}{|}^2\}{e}^{(\mathrm{\Phi }-2)\rho }dV.\end{array}$$

(3.13)

Next, it follows from the assumption that $m\geq 2\Phi -2$, and we apply Young’s inequality, then

$$\begin{array}{rl}& {\int }_{{\mathcal{N}}^m}\{\left(\frac{ϵ+3}{4}\right)+\left(\frac{ϵ-1}{4}\right)\left(\frac{2d_2+d_3(6{cos}^2\vartheta -4)}{m(m-1)}\right)+|\mathbb{H}{|}^2\}{e}^{(\mathrm{\Phi }-2)\rho }dV\\ & \le \frac{2}{\mathrm{\Phi }}{\int }_{{\mathcal{N}}^m}{\left\{\right|\left(\frac{ϵ+3}{4}\right)+\left(\frac{ϵ-1}{4}\right)\left(\frac{2d_2+d_3(6{cos}^2\vartheta -4)}{m(m-1)}\right)+|\mathbb{H}{|}^2|\}}^{\frac{\mathrm{\Phi }}{2}}dV\\ & +\frac{(\mathrm{\Phi }-2)}{\mathrm{\Phi }}{\int }_{{\mathcal{N}}^m}{e}^{\frac{\mathrm{\Phi }}{\rho }}dV.\end{array}$$

(3.14)

From (3.13) and (3.14) we deduce the following inequality:

$${\int }_{{\mathcal{N}}^m}{e}^{p\rho }dV\le {\int }_{{\mathcal{N}}^m}{\left(\right|\left(\frac{ϵ+3}{4}\right)+\left(\frac{ϵ-1}{4}\right)\left(\frac{2d_2+d_3(6{cos}^2\vartheta -4)}{m(m-1)}\right)+|\mathbb{H}{|}^2|)}^{\frac{\mathrm{\Phi }}{2}}dV.$$

(3.15)

Now, putting (3.15) into (3.2) we obtain (1.8). In the case of slant submanifolds, the equality case holds in (1.7), then considering the cases in (3.3) and (3.5), we obtain:

$$\begin{aligned} & \bigl\vert \varpi ^{a} \bigr\vert ^{2}= \bigl\vert \varpi ^{a} \bigr\vert ^{\Phi}, \\ &\Delta _{\Phi}\varpi ^{a}=-\Lambda _{1,\Phi} \bigl\vert \varpi ^{a} \bigr\vert ^{\Phi -2} \varpi ^{a}, \end{aligned}$$

for each $a=1,\ldots , 2k+2$. If $1<\Phi <2$ then $|\varpi ^{a}|=0$ or 1. Hence, there would be only one a for which $|\varpi ^{a}|=1$ and $\Lambda _{1,\Phi}=0$, which seems to be a contradiction as the eigenvalue is nonzero. Hence, we consider $\Phi =2$ and we are only restricted to the Laplacian case. Then, we are able to apply Theorem 1.5 from [15].

Let $\Phi >2$ and the equality remains valid in (1.8), then it shows that (3.7) and (3.8) become the equalities that indicates

$$\begin{aligned} \bigl\vert \varpi ^{1} \bigr\vert ^{\Phi}=\cdots = \bigl\vert \varpi ^{2k+2} \bigr\vert ^{\Phi } \end{aligned}$$

and condition $|\nabla \varpi ^{a}|=0$ holds for existing a. This shows that $\varpi ^{a}$ is a constant value and $\Lambda _{1, \Phi}$ is also equal to zero. This last result again represents a conflict in that $\Lambda _{1, \Phi}$ is a nonnull eigenvalue. This completes the proof of the theorem.

3.2 Proof of Theorem 1.2

Suppose that $1<\Phi \leq 2$, we have $\frac{\Phi}{2(\Phi -1)}\geq 1$. Then, by the Hölder inequality, we have:

$$\begin{array}{rl}& {\int }_{{\mathcal{N}}^m}\{\left(\frac{ϵ+3}{4}\right)+\left(\frac{ϵ-1}{4}\right)\left(\frac{2d_2+d_3(6{cos}^2\vartheta -4)}{m(m-1)}\right)+|\mathbb{H}{|}^2\}dV\\ & \le \left({(Vol(\mathcal{N}))}^{1-\frac{2(\mathrm{\Phi }-1)}{\mathrm{\Phi }}}\right)\\ & \times {\left\{{\int }_{{\mathcal{N}}^m}{\{\left(\frac{ϵ+3}{4}\right)+\left(\frac{ϵ-1}{4}\right)\left(\frac{2d_2+d_3(6{cos}^2\vartheta -4)}{m(m-1)}\right)+|\mathbb{H}{|}^2\}}^{\frac{\mathrm{\Phi }}{2(\mathrm{\Phi }-1)}}dV\right\}}^{\frac{2(\mathrm{\Phi }-1)}{\mathrm{\Phi }}}.\end{array}$$

(3.16)

Thus, combining the equations (1.7) with (3.16), we obtain the desired result (1.10). This completes the proof of the theorem.

Remark 3.2

As a result of the observations in Remark 3.1, the next result will be specified as a special version of Theorem 1.1. To be precise, we determine the following result by replacing $\epsilon =1$ in (1.7) and (1.8), respectively.

Corollary 3.1

Assume $\mathcal{N}^{m}$ is an $m(\geq 2)$-dimensional closed orientated semislant submanifold in $(2k+1)$-sphere ${\mathbb{S}}^{2k+1}(1)$, then, $\Lambda _{1, \Phi}$ satisfies the following inequality for the Φ-Laplacian

$$\begin{array}{rl}{\mathrm{\Lambda }}_{1,\mathrm{\Phi }}& \le \frac{2^{1-\frac{\mathrm{\Phi }}{2}}{(k+1)}^{(1-\frac{\mathrm{\Phi }}{2})}{m}^{\frac{\mathrm{\Phi }}{2}}}{{(Vol(\mathcal{N}))}^{p/2}}{\left\{{\int }_{{\mathcal{N}}^m}(1+|\mathbb{H}{|}^2)dV\right\}}^{\frac{\mathrm{\Phi }}{2}}\\ & \mathit{\text{for }}1<\mathrm{\Phi }\le 2,\end{array}$$

(3.17)

and

$$\begin{array}{rl}{\mathrm{\Lambda }}_{1,\mathrm{\Phi }}& \le \frac{2^{1-\frac{\mathrm{\Phi }}{2}}{(k+1)}^{(1-\frac{\mathrm{\Phi }}{2})}{m}^{\frac{\mathrm{\Phi }}{2}}}{Vol(\mathcal{N})}{\int }_{{\mathcal{N}}^k}{\left(\right|1+|\mathbb{H}{|}^2|)}^{\frac{\mathrm{\Phi }}{2}}dV\\ & \mathit{\text{for }}2<\mathrm{\Phi }\le \frac{m}{2}+1.\end{array}$$

(3.18)

There is another corollary based on Corollary 1.2 and it is as follows:

Corollary 3.2

Assuming that $\mathcal{N}^{m}$ is an $m(\geq 2)$-dimensional closed orientated semislant submanifold in $(2k+1)$-sphere ${\mathbb{S}}^{2k+1}(1)$, then, $\Lambda _{1, \Phi}$ satisfies the following inequality for the Φ-Laplacian

$${\mathrm{\Lambda }}_{1,\mathrm{\Phi }}\le \frac{{(2k+2)}^{(1-\frac{\mathrm{\Phi }}{2})}{m}^{\frac{\mathrm{\Phi }}{2}}}{{(Vol(\mathcal{N}))}^{(\mathrm{\Phi }-1)}}{\left\{{\int }_{{\mathcal{N}}^m}{(1+|\mathbb{H}{|}^2)}^{\frac{\mathrm{\Phi }}{2(\mathrm{\Phi }-1)}}dV\right\}}^{(\mathrm{\Phi }-1)}$$

(3.19)

for $1<\Phi \leq 2$.

3.3 Application to slant submanifolds of Sasakian space forms

Using Remark 2.1 and Theorem 1.1, we have the following results:

Corollary 3.3

([18])

Let $\mathcal{N}^{m}$ be an $m(\geq 2)$-dimensional closed orientated slant submanifold in a Sasakian space form ${\tilde{\mathbb{M}}}^{2k+1}(ϵ)$, then, $\Lambda _{1, \Phi}$ satisfies the following inequality for the Φ-Laplacian

$$\begin{array}{rl}{\mathrm{\Lambda }}_{1,\mathrm{\Phi }}& \le \left(\frac{2^{(1-\frac{\mathrm{\Phi }}{2})}{(k+1)}^{(1-\frac{\mathrm{\Phi }}{2})}{m}^{\frac{\mathrm{\Phi }}{2}}}{{(Vol(\mathcal{N}))}^{\mathrm{\Phi }/2}}\right)\\ & \times {\left\{{\int }_{{\mathcal{N}}^m}\{\left(\frac{ϵ+3}{4}\right)+\left(\frac{ϵ-1}{4}\right)\left(\frac{3{cos}^2\vartheta -2}{m}\right)+|\mathbb{H}{|}^2\}dV\right\}}^{\mathrm{\Phi }/2}\\ & \mathit{\text{for }}1<\mathrm{\Phi }\le 2,\end{array}$$

(3.20)

and

$$\begin{array}{rl}{\mathrm{\Lambda }}_{1,\mathrm{\Phi }}& \le \left(\frac{2^{(1-\frac{\mathrm{\Phi }}{2})}{(k+1)}^{(\frac{\mathrm{\Phi }}{2}-1)}{m}^{\frac{\mathrm{\Phi }}{2}}}{Vol(\mathcal{N})}\right)\\ & \times {\int }_{{\mathcal{N}}^m}{\left\{\right|\left(\frac{ϵ+3}{4}\right)+\left(\frac{ϵ-1}{4}\right)\left(\frac{3{cos}^2\vartheta -2}{m}\right)+|\mathbb{H}{|}^2|\}}^{\mathrm{\Phi }/2}dV\\ & \mathit{\text{for }}2<\mathrm{\Phi }\le \frac{m}{2}+1.\end{array}$$

(3.21)

From Corollary 1.1 for $\Phi =2$ and Remark 2.1, we have

Corollary 3.4

([18])

Assuming that $\mathcal{N}^{m}$ is an m-dimensional closed orientated slant submanifold in a Sasakian space form ${\tilde{\mathbb{M}}}^{2k+1}(ϵ)$, then, $\Lambda _{1}^{\Delta}$ satisfies the following inequality for the Laplacian

$${\mathrm{\Lambda }}_1^{\mathrm{\Delta }}\le \frac{m}{Vol(\mathcal{N})}{\int }_{\mathcal{N}}\{|\mathbb{H}{|}^2+\left(\frac{ϵ+3}{4}\right)+\left(\frac{ϵ-1}{4}\right)\left(\frac{3{cos}^2\vartheta -2}{m}\right)\}dV.$$

(3.22)

Similarly, from Theorem 1.2, we find that:

Corollary 3.5

([18])

Assuming that $\mathcal{N}^{m}$ is an $m(\geq 2)$-dimensional closed orientated slant submanifold in a Sasakian space form ${\tilde{\mathbb{M}}}^{2k+1}(ϵ)$, then $\Lambda _{1, \Phi}$ satisfies the following inequality for the Φ-Laplacian

$$\begin{array}{rl}{\mathrm{\Lambda }}_{1,\mathrm{\Phi }}& \le \left(\frac{2^{1-\frac{\mathrm{\Phi }}{2}}{(k+1)}^{(1-\frac{\mathrm{\Phi }}{2})}{m}^{\frac{\mathrm{\Phi }}{2}}}{{(Vol(\mathcal{N}))}^{(\mathrm{\Phi }-1)}}\right)\\ & \times {\left\{{\int }_{{\mathcal{N}}^m}{(\left(\frac{ϵ+3}{4}\right)+\left(\frac{ϵ-1}{4}\right)\left(\frac{3{cos}^2\vartheta -2}{m}\right)+|\mathbb{H}{|}^2)}^{\frac{\mathrm{\Phi }}{2(\mathrm{\Phi }-1)}}dV\right\}}^{(\mathrm{\Phi }-1)}\end{array}$$

(3.23)

for $1<\Phi \leq 2$.

3.4 Application to antiinvariant submanifolds of Sasakian space forms

Using Remark 2.2 and Theorem 1.1, we have the following results:

Corollary 3.6

Let $\mathcal{N}^{m}$ be an $m(\geq 2)$-dimensional closed orientated antiinvariant submanifold in a Sasakian space form ${\tilde{\mathbb{M}}}^{2k+1}(ϵ)$, then $\Lambda _{1, \Phi}$ satisfies the following inequality for the Φ-Laplacian

$$\begin{array}{rl}{\mathrm{\Lambda }}_{1,\mathrm{\Phi }}& \le \left(\frac{2^{(1-\frac{\mathrm{\Phi }}{2})}{(k+1)}^{(1-\frac{\mathrm{\Phi }}{2})}{m}^{\frac{\mathrm{\Phi }}{2}}}{{(Vol(\mathcal{N}))}^{\mathrm{\Phi }/2}}\right){\left\{{\int }_{{\mathcal{N}}^m}\{\left(\frac{ϵ+3}{4}\right)-\left(\frac{ϵ-1}{2m}\right)+|\mathbb{H}{|}^2\}dV\right\}}^{p/2}\\ & \mathit{\text{for }}1<\mathrm{\Phi }\le 2,\end{array}$$

(3.24)

and

$$\begin{array}{rl}{\mathrm{\Lambda }}_{1,\mathrm{\Phi }}& \le \left(\frac{2^{(1-\frac{\mathrm{\Phi }}{2})}{(k+1)}^{(\frac{\mathrm{\Phi }}{2}-1)}{m}^{\frac{\mathrm{\Phi }}{2}}}{Vol(\mathcal{N})}\right){\int }_{{\mathcal{N}}^m}{\left\{\right|\left(\frac{ϵ+3}{4}\right)-\left(\frac{ϵ-1}{2m}\right)+|\mathbb{H}{|}^2|\}}^{\mathrm{\Phi }/2}dV\\ & \mathit{\text{for }}2<\mathrm{\Phi }\le \frac{m}{2}+1.\end{array}$$

(3.25)

From Corollary 1.1 for $\Phi =2$ and Remark 2.2, we have:

Corollary 3.7

Assuming that $\mathcal{N}^{m}$ is an m-dimensional closed orientated antiinvariant submanifold in a Sasakian space form ${\tilde{\mathbb{M}}}^{2k+1}(ϵ)$, then $\Lambda _{1}^{\Delta}$ satisfies the following inequality for the Laplacian

$${\mathrm{\Lambda }}_1^{\mathrm{\Delta }}\le \frac{m}{Vol(\mathcal{N})}{\int }_{\mathcal{N}}\{|\mathbb{H}{|}^2+\left(\frac{ϵ+3}{4}\right)-\left(\frac{ϵ-1}{2m}\right)\}dV.$$

(3.26)

Similarly, from Theorem 1.2, we find that:

Corollary 3.8

Assuming that $\mathcal{N}^{m}$ is an $m(\geq 2)$-dimensional closed orientated antiinvariant submanifold in a Sasakian space form ${\tilde{\mathbb{M}}}^{2k+1}(ϵ)$, then $\Lambda _{1, \Phi}$ satisfies the following inequality for the Φ-Laplacian

$${\mathrm{\Lambda }}_{1,\mathrm{\Phi }}\le \left(\frac{2^{1-\frac{\mathrm{\Phi }}{2}}{(k+1)}^{(1-\frac{\mathrm{\Phi }}{2})}{m}^{\frac{\mathrm{\Phi }}{2}}}{{(Vol(\mathcal{N}))}^{(\mathrm{\Phi }-1)}}\right){\left\{{\int }_{{\mathcal{N}}^m}{(\left(\frac{ϵ+3}{4}\right)-\left(\frac{ϵ-1}{2m}\right)+|\mathbb{H}{|}^2)}^{\frac{\mathrm{\Phi }}{2(\mathrm{\Phi }-1)}}dV\right\}}^{(\mathrm{\Phi }-1)}$$

(3.27)

for $1<\Phi \leq 2$.

3.5 Application to contact CR-submanifolds of Sasakian space forms

Corollary 3.9

Let $\mathcal{N}^{m}$ be an $m(\geq 2)$-dimensional closed orientated contact CR-submanifold in a Sasakian space form ${\tilde{\mathbb{M}}}^{2k+1}(ϵ)$, then $\Lambda _{1, \Phi}$ satisfies the following inequalities for the Φ-Laplacian

• (1) The first nonnull eigenvalue $\Lambda _{1,\Phi}$ of the Φ-Laplacian satisfies:

  $$\begin{array}{rl}{\mathrm{\Lambda }}_{1,\mathrm{\Phi }}& \le \left(\frac{2^{(1-\frac{\mathrm{\Phi }}{2})}{(k+1)}^{(1-\frac{\mathrm{\Phi }}{2})}{m}^{\frac{\mathrm{\Phi }}{2}}}{{(Vol(\mathcal{N}))}^{\mathrm{\Phi }/2}}\right)\\ & \times {\left\{{\int }_{{\mathcal{N}}^m}\{\left(\frac{ϵ+3}{4}\right)+\left(\frac{ϵ-1}{4}\right)\left(\frac{2d_2-4)}{m(m-1)}\right)+|\mathbb{H}{|}^2\}dV\right\}}^{\mathrm{\Phi }/2}\\ & \mathit{\text{for }}1<\mathrm{\Phi }\le 2,\end{array}$$

  (3.28)

  and

  $$\begin{array}{rl}{\mathrm{\Lambda }}_{1,\mathrm{\Phi }}& \le \left(\frac{2^{(1-\frac{\mathrm{\Phi }}{2})}{(k+1)}^{(\frac{\mathrm{\Phi }}{2}-1)}{m}^{\frac{\mathrm{\Phi }}{2}}}{Vol(\mathcal{N})}\right)\\ & \times {\int }_{{\mathcal{N}}^m}{\left\{\right|\left(\frac{ϵ+3}{4}\right)+\left(\frac{ϵ-1}{4}\right)\left(\frac{2d_2-4)}{m(m-1)}\right)+|\mathbb{H}{|}^2|\}}^{\mathrm{\Phi }/2}dV\\ & \mathit{\text{for }}2<\mathrm{\Phi }\le \frac{m}{2}+1.\end{array}$$

  (3.29)

• (2) The equality carries in (1.7) and (1.8) if and only if $\Phi =2$ and $\mathcal{N}^{m}$ is minimally immersed in a geodesic sphere of radius $r_{\epsilon}$ of ${\tilde{\mathbb{M}}}^{2k+1}(ϵ)$ with the following equalities

  $$\begin{aligned} &r_{0}= \biggl(\frac{m}{\Lambda _{1}^{\Delta}} \biggr)^{1/2}, \\ &r_{1}=\sin ^{-1}{r_{0}}, \\ &r_{-1}=\sinh ^{-1}{r_{0}}. \end{aligned}$$