Results regarding the argument of certain p-valent analytic functions defined by a generalized integral operator

Abstract

The integral operator $J_p^m(\lambda ,\ell )(\lambda >0;\ell \ge 0;p\in \mathbb{N};m\in {\mathbb{N}}_0=\mathbb{N}\cup \{0\}$, where ℕ = {1,2,...}) for functions of the form $f\left(z\right)=z^p+\sum _{k=p+1}^{\infty }{a}_k{z}^k$ which are analytic and p-valent in the open unit disc U = {z ∈ ℂ: |z| < 1} was introduced by El-Ashwah and Aouf. The object of the present article is to drive interesting argument results of p-valent analytic functions defined by this integral operator.

2010 Mathematics Subject Classification: 30C45.

1 Introduction

Let A(p) denotes the class of functions of the form:

$$f\left(z\right)=z^p+\sum _{k=p+1}^{\infty }{a}_k{z}^k\left(p\in \mathbb{N}=\left\{1,2,\dots \right\}\right),$$

(1.1)

which are analytic and p-valent in the open unit disc U = {z ∈ ℂ: |z| < 1}. We note that A(1) = A, the class of univalent functions.

In [1], Catas defined the linear operator $J_p^m\left(\lambda ,\ell \right)f\left(z\right)$ as follows:

$$\begin{array}{c}{I}_p^m\left(\lambda ,\ell \right)f\left(z\right)=z^p+\sum _{k=p+1}^{\infty }{\left(\frac{p+\ell +\lambda \left(k-p\right)}{p+\ell }\right)}^m{a}_k{z}^k\\ \left(\lambda \ge 0;\ell \ge 0;p\in \mathbb{N};m\in {\mathbb{N}}_0\right).\end{array}$$

(1.2)

Also, El-Ashwah and Aouf [2] defined the integral operator $J_p^m\left(\lambda ,\ell \right)f\left(z\right)$ as follows:

$$\begin{array}{c}{J}_p^m\left(\lambda ,\ell \right)f\left(z\right)=z^p+\sum _{k=p+1}^{\infty }{\left(\frac{p+\ell }{p+\ell +\lambda \left(k-p\right)}\right)}^m{a}_k{z}^k\\ \left(\lambda \ge 0;\ell \ge 0;p\in \mathbb{N};m\in {\mathbb{N}}_0\right).\end{array}$$

(1.3)

The operator $J_p^m\left(\lambda ,\ell \right)f\left(z\right)$ was studied by Srivastava et al. [3] and Aouf et al. [4].

From (1.2) and (1.3), we observe that $J_p^{-m}\left(\lambda ,\ell \right)f\left(z\right)=I_p^m\left(\lambda ,\ell \right)f\left(z\right)\left(m>0\right)$, so the operator $J_p^m\left(\lambda ,\ell \right)f\left(z\right)$ is well-defined for λ ≥ 0, ℓ ≥ 0, p ∈ ℕ and m ∈ ℤ = {..., -2, -1,0,1,2,...}.

From (1.3), it is easy to verify that (see, [2])

$$\begin{array}{c}\lambda z{\left(J_p^{m+1}\left(\lambda ,\ell \right)f\left(z\right)\right)}^{\prime }=\left(\ell +p\right)J_p^m\left(\lambda ,\ell \right)f\left(z\right)-\left[\ell +p\left(1-\lambda \right)\right]J_p^{m+1}\left(\lambda ,\ell \right)f\left(z\right)\\ \left(\lambda >0;\ell \ge 0;p\in \mathbb{N};m\in {\mathbb{N}}_0\right).\end{array}$$

(1.4)

We note that:

• (i) $J_1^m\left(\lambda ,0\right)f\left(z\right)=I_{\lambda }^{-m}f\left(z\right)\left(m\ge 0\right)$ (see Patel [5]);

• (ii) $J_p^{\alpha }\left(1,1\right)f\left(z\right)=I_p^{\alpha }f\left(z\right)\left(\alpha >0\right)$ (see Shams et al. [6]);

• (iii) $J_p^m\left(1,1\right)f\left(z\right)=D^{m}f\left(z\right)$ (see Patel and Sahoo [7]);

• (iv) $J_1^m\left(\lambda ,0\right)f\left(z\right)=I_{\lambda }^{m}f\left(z\right)$ (see Al-Oboudi and Al-Qahtani [8]);

• (v) $J_1^{\alpha }\left(1,1\right)f\left(z\right)=I^{\alpha }f\left(z\right)\left(\alpha >0\right)$ (see Jung et al. [9]);

• (vi) $J_1^m\left(1,1\right)f\left(z\right)=I^{m}f\left(z\right)$ (see Flett [10]);

• (vii) $J_1^m\left(1,0\right)f\left(z\right)={ℒ}^{m}f\left(z\right)$ (see, Salagean [11]).

Also we note that:

• (i) $J_p^m\left(1,0\right)f\left(z\right)=J_p^{m}f\left(z\right)=z^p+\sum _{k=p+1}^{\infty }{\left(\frac{p}{k}\right)}^m{a}_k{z}^k\left(p\in \mathbb{N};m\in {\mathbb{N}}_0;z\in U\right)$;

• (ii) $J_p^m\left(1,\ell \right)f\left(z\right)=J_p^m\left(\ell \right)f\left(z\right)=z^p+\sum _{k=p+1}^{\infty }{\left(\frac{p+\ell }{k+\ell }\right)}^m{a}_k{z}^k$

  $$\left(p\in \mathbb{N};m\in {\mathbb{N}}_0;\ell \ge 0;z\in U\right).$$

In this article, we drive interesting argument results of p-valent analytic functions defined by the integral operator $J_p^m\left(\lambda ,\ell \right)f\left(z\right)$.

2 Main results

In order to prove our main results, we recall the following lemma.

Lemma 1[12]. Let h(z) be analytic in U with h(0) ≠ 0 (z ∈ U). Further suppose that α, β ∈ ℝ⁺ = (0, ∞) and

$$\left|\text{arg}\left(h\left(z\right)+\beta z{h}^{\prime }\left(z\right)\right)\right|<\frac{\pi }{2}\left(\alpha +\frac{2}{\pi }\text{arctan}\left(\beta \alpha \right)\right)\left(\alpha ,\beta >0\right),$$

(2.1)

then

$$\left|\text{arg}\left(h\left(z\right)\right)\right|<\frac{\pi }{2}\alpha \left(z\in U\right).$$

(2.2)

Unless otherwise mentioned we shall assume throughout the article that α, γ, δ ∈ ℝ⁺, λ > 0, ℓ ≥ 0, p ∈ ℕ, m ∈ ℤ and the powers are understood as principle values.

Theorem 1. Let g(z) ∈ A(p). Suppose f(z) ∈ A(p) satisfies the following condition

$$\begin{array}{l}\left|\text{arg}\left({\left\{\frac{J_p^m\left(\lambda ,\ell \right)f\left(z\right)}{J_p^m\left(\lambda ,\ell \right)g\left(z\right)}\right\}}^{\gamma }\left\{1+\frac{\delta }{\lambda }\left(\frac{J_p^{m-1}\left(\lambda ,\ell \right)f\left(z\right)}{J_p^m\left(\lambda ,\ell \right)f\left(z\right)}-\frac{J_p^{m-1}\left(\lambda ,\ell \right)g\left(z\right)}{J_p^m\left(\lambda ,\ell \right)g\left(z\right)}\right)\right\}\right)\right|\\ <\frac{\pi }{2}\left(\alpha +\frac{2}{\pi }\text{arctan}\left[\frac{\delta }{\gamma \left(\ell +p\right)}\alpha \right]\right)\end{array}$$

(2.3)

then

$$\left|\text{arg}{\left\{\frac{J_p^m\left(\lambda ,\ell \right)f\left(z\right)}{J_p^m\left(\lambda ,\ell \right)g\left(z\right)}\right\}}^{\gamma }\right|<\frac{\pi }{2}\alpha \left(z\in U\right).$$

(2.4)

Proof. Define a function

$$h\left(z\right)={\left\{\frac{J_p^m\left(\lambda ,\ell \right)f\left(z\right)}{J_p^m\left(\lambda ,\ell \right)g\left(z\right)}\right\}}^{\gamma },\gamma \ne 0$$

(2.5)

then h(z) = 1 + c₁z + ⋯, is analytic in U with h(0) = 1 and h'(0) ≠ 0.

Differentiating (2.5) logarithmically with respect to z and multiplying by z, we have

$$\frac{1}{\gamma }\frac{z{h}^{\prime }\left(z\right)}{h\left(z\right)}=\left\{\frac{z{\left(J_p^m\left(\lambda ,\ell \right)f\left(z\right)\right)}^{\prime }}{J_p^m\left(\lambda ,\ell \right)f\left(z\right)}-\frac{z{\left(J_p^m\left(\lambda ,\ell \right)g\left(z\right)\right)}^{\prime }}{J_p^m\left(\lambda ,\ell \right)g\left(z\right)}\right\}.$$

(2.6)

Using (1.4) in (2.6), we obtain

$$h\left(z\right)+\frac{\delta }{\gamma \left(\ell +p\right)}z{h}^{\prime }\left(z\right)={\left\{\frac{J_p^m\left(\lambda ,\ell \right)f\left(z\right)}{J_p^m\left(\lambda ,\ell \right)g\left(z\right)}\right\}}^{\gamma }\left\{1+\frac{\delta }{\lambda }\left(\frac{J_p^{m-1}\left(\lambda ,\ell \right)f\left(z\right)}{J_p^m\left(\lambda ,\ell \right)f\left(z\right)}-\frac{J_p^{m-1}\left(\lambda ,\ell \right)g\left(z\right)}{J_p^m\left(\lambda ,\ell \right)g\left(z\right)}\right)\right\}.$$

(2.7)

By using Lemma 1, the proof of Theorem 1 is completed.

Remark 1. Putting λ = δ = p = 1, ℓ = m = 0, and g(z) = z, in Theorem 1, we obtain the result obtained by Lashin [12, Theorem 2.2].

Putting γ = 1 and g(z) = z^pin Theorem 1, we obtain the following corollary:

Corollary 1. If f(z) ∈ A(p) satisfies the following condition

$$\left|\text{arg}\left\{\frac{\delta }{\lambda }\frac{J_p^{m-1}\left(\lambda ,\ell \right)f\left(z\right)}{z^p}+\left(1-\frac{\delta }{\lambda }\right)\frac{J_p^m\left(\lambda ,\ell \right)f\left(z\right)}{z^p}\right\}\right|<\frac{\pi }{2}\left(\alpha +\frac{2}{\pi }\text{arctan}\left[\frac{\delta }{\left(\ell +p\right)}\alpha \right]\right)$$

(2.8)

then

$$\left|\text{arg}\left(\frac{J_p^m\left(\lambda ,\ell \right)f\left(z\right)}{z^p}\right)\right|<\frac{\pi }{2}\alpha \left(z\in U\right).$$

(2.9)

Next, putting p = 1 in Corollary 1, we obtain the following corollary:

Corollary 2. If f(z) ∈ A satisfies the following condition

$$\left|\text{arg}\left\{\frac{\delta }{\lambda }\frac{J_1^{m-1}\left(\lambda ,\ell \right)f\left(z\right)}{z}+\left(1-\frac{\delta }{\lambda }\right)\frac{J_1^m\left(\lambda ,\ell \right)f\left(z\right)}{z}\right\}\right|<\frac{\pi }{2}\left(\alpha +\frac{2}{\pi }\text{arctan}\left[\frac{\delta }{\left(\ell +1\right)}\alpha \right]\right)$$

(2.10)

then

$$\left|\text{arg}\left(\frac{J_1^m\left(\lambda ,\ell \right)f\left(z\right)}{z}\right)\right|<\frac{\pi }{2}\alpha \left(z\in U\right).$$

(2.11)

Remark 2. Putting λ = 1 and ℓ = m = 0 in Corollary 2 we obtain the result obtained by Lashin [12, Example 2.2].

Finally, putting γ = 1 and f(z) = z^pin Theorem 1, we obtain the following corollary:

Corollary 3. Let $\frac{z^p}{J_p^m\left(\lambda ,\ell \right)g\left(z\right)}\ne 0,g\left(z\right)\in A\left(p\right)$ and δ ≥ 0. Suppose that

$$\begin{array}{l}\left|\text{arg}\left\{\left(1+\frac{\delta }{\lambda }\right)\frac{z^p}{J_p^m\left(\lambda ,\ell \right)g\left(z\right)}-\frac{\delta }{\lambda }\frac{J_p^{m-1}\left(\lambda ,\ell \right)g\left(z\right)}{J_p^m\left(\lambda ,\ell \right)g\left(z\right)}\left(\frac{z^p}{J_p^m\left(\lambda ,\ell \right)g\left(z\right)}\right)\right\}\right|\\ <\frac{\pi }{2}\left(\alpha +\frac{2}{\pi }\text{arctan}\left[\frac{\delta }{\left(\ell +p\right)}\alpha \right]\right)\end{array}$$

(2.12)

then

$$\left|\text{arg}\left(\frac{z^p}{J_p^m\left(\lambda ,\ell \right)g\left(z\right)}\right)\right|<\frac{\pi }{2}\alpha .$$

(2.13)

Theorem 2. Let 0 < δ ≤ 1. Suppose f(z) ∈ A(p) satisfies the following condition

$$\left|\text{arg}\left(\frac{J_p^m\left(\lambda ,\ell \right)f\left(z\right)}{z^p}\right)\right|<\frac{\pi }{2}\left(\alpha +\frac{2}{\pi }\text{arctan}\left[\frac{\delta }{\gamma \left(\ell +p\right)}\alpha \right]\right)\left(z\in U\right)$$

(2.14)

then we have

$$\left|\text{arg}\left(\frac{\gamma \left(\ell +p\right)}{\delta }{z}^{-\frac{\gamma \left(\ell +p\right)}{\delta }}\underset{0}{\overset{z}{\int }}{t}^{\frac{\gamma \left(\ell +p\right)-\delta \left(p+1\right)}{\delta }}{J}_p^m\left(\lambda ,\ell \right)f\left(t\right)dt\right)\right|<\frac{\pi }{2}\alpha .$$

(2.15)

Proof. Consider the function

$$h\left(z\right)=\frac{\gamma \left(\ell +p\right)}{\delta }{z}^{-\frac{\gamma \left(\ell +p\right)}{\delta }}\underset{0}{\overset{z}{\int }}{t}^{\frac{\gamma \left(\ell +p\right)-\delta \left(p+1\right)}{\delta }}{J}_p^m\left(\lambda ,\ell \right)f\left(t\right)dt$$

(2.16)

then h(z) = 1 + c₁z + ⋯, is analytic in U with h(0) = 1 and h'(0) ≠ 0.

Differentiating (2.16) with respect to z, we have

$$h\left(z\right)+\frac{\delta }{\gamma \left(\ell +p\right)}z{h}^{\prime }\left(z\right)=\frac{J_p^m\left(\lambda ,\ell \right)f\left(z\right)}{z^p}.$$

(2.17)

By using Lemma 1, the proof of Theorem 2 is completed.

Putting p = δ = γ = 1 and m = 0 in Theorem 2 we obtain the following corollary:

Corollary 4. Let f(z) ∈ A satisfies the following condition

$$\left|\text{arg}\left(\frac{f\left(z\right)}{z}\right)\right|<\frac{\pi }{2}\left(\alpha +\frac{2}{\pi }\text{arctan}\left[\frac{\alpha }{\ell +1}\right]\right)$$

(2.18)

then

$$\left|\text{arg}\left(\frac{\ell +1}{z^{\left(\ell +1\right)}}\underset{0}{\overset{z}{\int }}{t}^{\left(\ell -1\right)}f\left(t\right)dt\right)\right|<\frac{\pi }{2}\alpha \left(z\in U\right).$$

(2.19)

Remark 3. (i) Putting ℓ = 0 in Corollary 4 we obtain the result obtained by Goyal and Goswami [13, Corollary 3.6].

• (ii) By specifying the parameters p, λ, ℓ, and m we obtain various results for different operators reminded in the introduction.