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On extreme points and product properties of a new subclass of p-harmonic functions
Journal of Inequalities and Applications volume 2019, Article number: 133 (2019)
Abstract
In this paper, we introduce a new subclass of p-harmonic functions and investigate the univalence and sense-preserving, extreme points, distortion bounds, convex combination, neighborhoods of mappings belonging to the subclass. Relevant connections of the results presented here with the results of previous research are briefly indicated. Finally, we also prove new properties of the Hadamard product of these classes.
1 Introduction
Let \(\mathcal{H}\) denote the class of all complex-valued harmonic functions \(f = h+\bar{g} \) in \(\mathbb{U}=\{z:|z|<1\}\), where h and g are analytic in \(\mathbb{U}\) and normalized such that
A necessary and sufficient condition for f to be locally univalent and sense-preserving in \(\mathbb{U}\) is that \(J_{f}=|f_{z}|^{2}-|f _{\bar{z}}|^{2}>0\) in \(\mathbb{U}\) (see [1, 2]). Let \(S_{H}\) denote the subclass of \(\mathcal{H}\) consisting of sense-preserving univalent functions in \(\mathbb{U}\). Then the function \(f \in S_{H} \) of the form (1.1) satisfies the condition \(|b_{1}|<1\).
A 2p-times continuously differentiable complex-valued function \(F =u+iv \) in a domain \(\mathbb{U}\) is p-harmonic if F satisfies the p-harmonic equation \(\bigtriangleup ^{p}F=\bigtriangleup ( \bigtriangleup ^{p-1}F)=0\ (p=1,2,\ldots )\), where Δ represents the complex Laplacian operator:
Obviously, if we take \(p=1\) and \(p=2\), then F is harmonic and biharmonic, respectively.
A function F is p-harmonic in a simply connected domain \(\mathbb{U}\) if and only if F has the following representation:
where each \(f_{p-k+1}(z)\) is harmonic (or \(\triangle f_{p-k+1}=0\)) (see [3]) and \(f_{p-k+1}(z)\) has the form
where
Denote by \(SH_{p}\) the class of functions F of the form (1.2) that are p-harmonic, univalent, and sense-preserving in the unit disk. Recently, there has been significant interest in results about the class \(SH_{p}\) (see, for details, [4,5,6,7,8,9]).
Denote by \(\mathit{HL}_{p}(\alpha,\lambda )\ (0\leq \alpha <1,\lambda \geq 0)\) the class of all mappings of the form (1.2) which satisfy the condition
with
Clearly, inequality (1.6) implies that
where \(a_{1,p}=1,k\in \{1,\ldots,p\}\).
It is easy to see that various subclasses of \(SH_{p}\) consisting of mappings \(F(z)\) of the form (1.2) and (1.3) can be represented as \(\mathit{HL}_{p}(\alpha,\lambda )(b_{1,p}=a_{1,p-k+1}=b_{1,p-k+1}=0,k=2, \ldots,p)\) for suitable choices of \(p,\alpha \), and λ in the earlier studies by various authors.
-
(i)
\(\mathit{HL}_{p}(0,0)=\mathit{HS}_{p}\) and \(\mathit{HL}_{p}(0,1)=\mathit{HC}_{p}\) (see Qiao and Wang [4]);
-
(ii)
\(\mathit{HL}_{p}(\alpha,0)=\mathit{HS}_{p}(\alpha )\) and \(\mathit{HL}_{p}(\alpha,1)=\mathit{HC} _{p}(\alpha )\) (see Saurabh Porwal and Dixit [5]);
-
(iii)
\(\mathit{HL}_{1}(\alpha,0)=\mathit{HS}(\alpha )\) and \(\mathit{HL}_{1}(\alpha,1)=\mathit{HC}( \alpha )\) (see Öztürk and Yalcin [10]);
-
(vi)
\(\mathit{HL}_{1}(0,0)=\mathit{HS}\) and \(\mathit{HL}_{1}(0,1)=\mathit{HC}\) (see Avci and Zlotkiewicz [11]).
For \(\lambda \in \mathbb{N}=\{1,2,\ldots \}\cup \{0\}\), we have the following inclusion relation:
Suppose that F is a p-harmonic mapping with expression (1.2). Following Ruscheweyh [12], we use \(N_{\lambda,\alpha }^{\delta }(F)\) to denote the δ-neighborhood of F in p-harmonic mappings:
where
If \(F, G\in SH_{p}\) satisfy
and
then the convolution \(F\ast G\) of F and G is defined to be the mapping
Let
and denote \(\overline{\mathit{HL}}_{p}(\alpha,\lambda )=\mathit{HL}_{p}(\alpha,\lambda )\cap \mathit{TH}_{p}\).
The main objective of the paper is to introduce a new subclass of p-harmonic mappings and investigate the univalence and sense-preserving, extreme points, neighborhoods and Hadamard product of mappings for the above subclass. Relevant connections of the results presented here with the results of Qiao et al. [4] and Porwal et al. [5] are briefly indicated. Finally, we also prove new properties of the Hadamard product of these classes.
2 Main results
Firstly, we discuss the inclusion relation of \(\mathit{HL}_{p}(\alpha,\lambda )\).
Theorem 2.1
Let \(\lambda _{2}\geq \lambda _{1}\geq 0, 1> \alpha _{2}\geq \alpha _{1}\geq 0\), then \(\mathit{HL}_{p}(\alpha _{2},\lambda _{2}) \subseteq \mathit{HL}_{p}(\alpha _{1},\lambda _{1})\).
Proof
Let \(F\in \mathit{HL}_{p}(\alpha _{2},\lambda _{2})\), then using (1.6), we have
therefore \(F\in \mathit{HL}_{p}(\alpha _{1},\lambda _{1})\), and so \(\mathit{HL}_{p}(\alpha _{2},\lambda _{2})\subseteq \mathit{HL}_{p}(\alpha _{1},\lambda _{1})\). □
Next, we prove that the mapping in \(\mathit{HL}_{p}(\alpha,\lambda )\) is univalent and sense-preserving.
Theorem 2.2
Each mapping in \(\mathit{HL}_{p}(\alpha,\lambda )\) is univalent and sense-preserving.
Proof
Let \(F\in \mathit{HL}_{p}(\alpha,\lambda )\) and \(z_{1},z _{2}\in \mathbb{U}\) with \(z_{1}\neq z_{2}\), so that \(|z_{1}|\leq |z _{2}|\):
which proves univalence.
In order to prove that F is sense-preserving, we need to show that \(J_{F}=|F_{z}|^{2}-|F_{\bar{z}}|^{2}>0\):
From \(z\neq 0\) and the obvious fact \(J_{F}(0)>0\), we thus complete the proof. □
Example 2.1
Let \(F(z)=z+\frac{1}{(2p-1)}|z|^{2(p-1)} \bar{z}\). Then \(F(z)\) is a p-harmonic function and
using (1.8), we get \(F\in \mathit{HL}_{p}(0,\lambda )\).
Also, we determine the extreme points of \(\overline{\mathit{HL}}_{p}(\alpha, \lambda )\).
Theorem 2.3
Let F be given by (1.2). Then \(F\in \overline{\mathit{HL}}_{p}(\alpha,\lambda )\) if and only if
where
and
In particular, the extreme points of \(\overline{\mathit{HL}}_{p}(\alpha, \lambda )\) are \(\{h_{j,p-k+1}(z)\}\) and \(\{g_{j,p-k+1}(z)\}\), where \(j\geq 1\) and \(1\leq k \leq p\).
Proof
Since
and
we see that \(F\in \overline{\mathit{HL}}_{p}(\alpha,\lambda )\).
Conversely, assuming that \(F\in \overline{\mathit{HL}}_{p}(\alpha,\lambda )\) and setting
and
where \(X_{1,p}\geq 0\). Then, as required, we obtain
 □
Example 2.2
Let \(F(z)=z+\frac{1}{(2p-1)}|z|^{2(p-1)}z+ \frac{1}{(2p-1)}|z|^{2(p-1)}\bar{z}\). Then \(F(z)\) is a p-harmonic function, and using Theorem 2.3, we have \(F\in \overline{\mathit{HL}}_{p}(0, \lambda )\). Here, we give the figures for \(p=4\) and \(p=10\), respectively (see Fig. 1 and Fig. 2).
Theorem 2.4
Let F be given by (1.2) and \(F\in \overline{\mathit{HL}}_{p}(\alpha,\lambda )\). Then, for \(|z|=r<1\), we have
and
where
Proof
Let \(F\in \overline{\mathit{HL}}_{p}(\alpha,\lambda )\). Taking the absolute value of \(F(z)\), we have
and
 □
Corollary 2.5
Let F be given by (1.2)Â and \(F\in \overline{\mathit{HL}}_{p}(\alpha,\lambda )\). Then
where
and \(\psi _{j,k }(\lambda,\alpha )\) is given by (2.4).
Theorem 2.6
The class \(F\in \overline{\mathit{HL}}_{p}( \alpha,\lambda )\) is closed under combination.
Proof
For \(i=1,2,\ldots \) , let \(F_{i}\in \overline{\mathit{HL}} _{p}(\alpha,\lambda )\), where
Then, by (1.6) and (2.4), we get
For \(\sum_{i=1}^{\infty }t_{i}=1,0\leq t_{i}\leq 1\), the convex combination of \(F_{i}\) may be written as
Then, by (2.5), we obtain
Therefore, using (1.6), we obtain \(\sum_{i=1}^{\infty }t_{i}F_{i} \in \overline{\mathit{HL}}_{p}(\alpha,\lambda )\). □
Theorem 2.7
Let
belong to \(\overline{\mathit{HL}}_{p}(\alpha,\lambda _{2})\). If \(\lambda _{2}> \lambda _{1}\geq 0\) and
then \(N_{\lambda _{1},\alpha }^{\delta }(F_{1})\subset \mathit{HL}_{p}(\alpha, \lambda _{1})\), where
Proof
The δ-neighborhood of \(F_{1}\) is the set
where
If
then we have
Hence \(F_{2}\in \overline{\mathit{HL}}_{p}(\alpha,\lambda _{1})\). □
Remark 2.8
-
1.
If \(\alpha =0,\lambda =0\) and \(\alpha =0,\lambda =1 \), then Theorem 2.2, Theorem 2.4, and Theorem 2.7, respectively, coincide with Theorem 3.1, Theorem 4.3, Theorem 4.4, Lemma 4.1, and Theorem 5.1 in [4].
-
2.
If \(\lambda =0\) and \(\lambda =1\), then Theorem 2.2, Theorem 2.3, and Theorem 2.7, respectively, coincide with Theorem 3.1, Theorem 3.6, Theorem 3.7, and Theorem 4.1 in [5].
At last, we discuss the Hadamard product of \(\overline{\mathit{HL}}_{p}(\alpha ,\lambda )\).
Theorem 2.9
Let \(\lambda \geq 0,~0\leq \alpha <1,~p \in \{1,2,\ldots \}\). If \(F,~G\in \overline{\mathit{HL}}_{p}(\alpha,\lambda )\), then \(F\ast G \in \overline{\mathit{HL}}_{p}(\alpha,\lambda )\), where
Proof
Let \(F,G\in \overline{\mathit{HL}}_{p}(\alpha,\lambda )\), then, from (1.8), we know that, in order to prove \(F\ast G \in \overline{\mathit{HL}}_{p}(\alpha,\lambda )\), we need to show that
Since \(F,G\in \overline{\mathit{HL}}_{p}(\alpha,\lambda )\), using (1.8), we have
and
From (2.10) and (2.11), we obtain
and
Using the Cauchy–Schwarz inequations, from (2.12) and (2.13), we get
because
So from (2.14) and (2.15), we have
and hence
which implies that
In addition, if
that is,
then we obtain the conditions of satisfaction (2.9). Again, combining (2.17) and (2.18) with \(k=1\) and \(j=2\), we can get
which deduces condition (2.8). The proof is completed. □
Taking \(\lambda =0\) and \(\lambda =1\) in Theorem 2.9, respectively, we obtain the following corollaries.
Corollary 2.10
Let \(0\leq \alpha <1,2-\alpha \geq 2(1-\alpha )p^{2}(p\geq 1)\). If \(F,G\in \mathit{HS}_{p}(\alpha )\), then \(F\ast G\in \mathit{HS}_{p}(\alpha )\).
Corollary 2.11
Let \(0\leq \alpha <1,2-\alpha \geq (1-\alpha )p^{2}(p\geq 1)\). If \(F,G\in \mathit{HC}_{p}(\alpha )\), then \(F\ast G\in \mathit{HC}_{p}(\alpha )\).
3 Conclusions
In this paper, we mainly introduce a new subclass of p-harmonic mappings and investigate the univalence and sense-preserving, extreme points, distortion bounds, convex combination, neighborhoods of mappings belonging to the subclass. Relevant connections of the results presented here with the results of Qiao et al. [4] and Porwal et al. [5] are briefly indicated. Finally, we also prove new properties of the Hadamard product of these classes.
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Acknowledgements
We would like to thank the referees for their valuable comments, suggestions, and corrections.
Funding
This work was supported by the Inner Mongolia Autonomous Region key institutions of higher learning scientific research projects (No. NJZZ19209), the present investigation was supported by the Natural Science Foundation of China (No. 11561001), the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region (Grant No. NJYT-18-A14), and the Natural Science Foundation of Inner Mongolia of the People’s Republic of China (Grant No. 2018MS01026).
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Li, SH., Tang, H. & Niu, XM. On extreme points and product properties of a new subclass of p-harmonic functions. J Inequal Appl 2019, 133 (2019). https://doi.org/10.1186/s13660-019-2092-9
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DOI: https://doi.org/10.1186/s13660-019-2092-9