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

## Abstract

The integral operator ${J}_{p}^{m}\left(\lambda ,\ell \right)\left(\lambda >0;\ell \ge 0;p\in ℕ;m\in {ℕ}_{0}=ℕ\cup \left\{0\right\}$, where = {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 : |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:

$f\left(z\right)={z}^{p}+\sum _{k=p+1}^{\infty }{a}_{k}{z}^{k}\phantom{\rule{1em}{0ex}}\left(p\in ℕ=\left\{1,2,\dots \right\}\right),$
(1.1)

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

In , Catas defined the linear operator ${J}_{p}^{m}\left(\lambda ,\ell \right)f\left(z\right)$ as follows:

$\begin{array}{c}{I}_{p}^{m}\left(\lambda ,\ell \right)f\left(z\right)={z}^{p}+\sum _{k=p+1}^{\infty }{\left(\frac{p+\ell +\lambda \left(k-p\right)}{p+\ell }\right)}^{m}{a}_{k}{z}^{k}\\ \left(\lambda \ge 0;\ell \ge 0;p\in ℕ;m\in {ℕ}_{0}\right).\end{array}$
(1.2)

Also, El-Ashwah and Aouf  defined the integral operator ${J}_{p}^{m}\left(\lambda ,\ell \right)f\left(z\right)$ as follows:

$\begin{array}{c}{J}_{p}^{m}\left(\lambda ,\ell \right)f\left(z\right)={z}^{p}+\sum _{k=p+1}^{\infty }{\left(\frac{p+\ell }{p+\ell +\lambda \left(k-p\right)}\right)}^{m}{a}_{k}{z}^{k}\\ \left(\lambda \ge 0;\ell \ge 0;p\in ℕ;m\in {ℕ}_{0}\right).\end{array}$
(1.3)

The operator ${J}_{p}^{m}\left(\lambda ,\ell \right)f\left(z\right)$ was studied by Srivastava et al.  and Aouf et al. .

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 and m = {..., -2, -1,0,1,2,...}.

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

$\begin{array}{c}\lambda z{\left({J}_{p}^{m+1}\left(\lambda ,\ell \right)f\left(z\right)\right)}^{\prime }=\left(\ell +p\right){J}_{p}^{m}\left(\lambda ,\ell \right)f\left(z\right)-\left[\ell +p\left(1-\lambda \right)\right]{J}_{p}^{m+1}\left(\lambda ,\ell \right)f\left(z\right)\\ \left(\lambda >0;\ell \ge 0;p\in ℕ;m\in {ℕ}_{0}\right).\end{array}$
(1.4)

We note that:

1. (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 );

2. (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. );

3. (iii)

${J}_{p}^{m}\left(1,1\right)f\left(z\right)={D}^{m}f\left(z\right)$ (see Patel and Sahoo );

4. (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 );

5. (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. );

6. (vi)

${J}_{1}^{m}\left(1,1\right)f\left(z\right)={I}^{m}f\left(z\right)$ (see Flett );

7. (vii)

${J}_{1}^{m}\left(1,0\right)f\left(z\right)={ℒ}^{m}f\left(z\right)$ (see, Salagean ).

Also we note that:

1. (i)

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

2. (ii)

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

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

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. Let h(z) be analytic in U with h(0) ≠ 0 (z U). Further suppose that α, β + = (0, ∞) and

$\left|\text{arg}\left(h\left(z\right)+\beta z{h}^{\prime }\left(z\right)\right)\right|<\frac{\pi }{2}\left(\alpha +\frac{2}{\pi }\text{arctan}\left(\beta \alpha \right)\right)\phantom{\rule{1em}{0ex}}\left(\alpha ,\beta >0\right),$
(2.1)

then

$\left|\text{arg}\left(h\left(z\right)\right)\right|<\frac{\pi }{2}\alpha \phantom{\rule{1em}{0ex}}\left(z\in U\right).$
(2.2)

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

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

$\begin{array}{l}\left|\text{arg}\left({\left\{\frac{{J}_{p}^{m}\left(\lambda ,\ell \right)f\left(z\right)}{{J}_{p}^{m}\left(\lambda ,\ell \right)g\left(z\right)}\right\}}^{\gamma }\left\{1+\frac{\delta }{\lambda }\left(\frac{{J}_{p}^{m-1}\left(\lambda ,\ell \right)f\left(z\right)}{{J}_{p}^{m}\left(\lambda ,\ell \right)f\left(z\right)}-\frac{{J}_{p}^{m-1}\left(\lambda ,\ell \right)g\left(z\right)}{{J}_{p}^{m}\left(\lambda ,\ell \right)g\left(z\right)}\right)\right\}\right)\right|\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}<\frac{\pi }{2}\left(\alpha +\frac{2}{\pi }\text{arctan}\left[\frac{\delta }{\gamma \left(\ell +p\right)}\alpha \right]\right)\phantom{\rule{2em}{0ex}}\end{array}$
(2.3)

then

$\left|\text{arg}{\left\{\frac{{J}_{p}^{m}\left(\lambda ,\ell \right)f\left(z\right)}{{J}_{p}^{m}\left(\lambda ,\ell \right)g\left(z\right)}\right\}}^{\gamma }\right|<\frac{\pi }{2}\alpha \phantom{\rule{1em}{0ex}}\left(z\in U\right).$
(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, = 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) = zpin 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

$\left|\text{arg}\left(\frac{{J}_{p}^{m}\left(\lambda ,\ell \right)f\left(z\right)}{{z}^{p}}\right)\right|<\frac{\pi }{2}\alpha \phantom{\rule{1em}{0ex}}\left(z\in U\right).$
(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

$\left|\text{arg}\left(\frac{{J}_{1}^{m}\left(\lambda ,\ell \right)f\left(z\right)}{z}\right)\right|<\frac{\pi }{2}\alpha \phantom{\rule{1em}{0ex}}\left(z\in U\right).$
(2.11)

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

Finally, putting γ = 1 and f(z) = zpin 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

$\begin{array}{l}\left|\text{arg}\left\{\left(1+\frac{\delta }{\lambda }\right)\frac{{z}^{p}}{{J}_{p}^{m}\left(\lambda ,\ell \right)g\left(z\right)}-\frac{\delta }{\lambda }\frac{{J}_{p}^{m-1}\left(\lambda ,\ell \right)g\left(z\right)}{{J}_{p}^{m}\left(\lambda ,\ell \right)g\left(z\right)}\left(\frac{{z}^{p}}{{J}_{p}^{m}\left(\lambda ,\ell \right)g\left(z\right)}\right)\right\}\right|\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}<\frac{\pi }{2}\left(\alpha +\frac{2}{\pi }\text{arctan}\left[\frac{\delta }{\left(\ell +p\right)}\alpha \right]\right)\phantom{\rule{2em}{0ex}}\end{array}$
(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

$\left|\text{arg}\left(\frac{{J}_{p}^{m}\left(\lambda ,\ell \right)f\left(z\right)}{{z}^{p}}\right)\right|<\frac{\pi }{2}\left(\alpha +\frac{2}{\pi }\text{arctan}\left[\frac{\delta }{\gamma \left(\ell +p\right)}\alpha \right]\right)\phantom{\rule{1em}{0ex}}\left(z\in U\right)$
(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 = 0 in Corollary 4 we obtain the result obtained by Goyal and Goswami [13, Corollary 3.6].

1. (ii)

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

## References

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## Acknowledgements

The author thanks the referees for their valuable suggestions which led to improvement of this study. Also, he would like to express his sincere gratitude to Springer Open Accounts Team for their kind help.

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Correspondence to R M El-Ashwah.

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The authors declare that they have no competing interests.

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El-Ashwah, R.M. Results regarding the argument of certain p-valent analytic functions defined by a generalized integral operator. J Inequal Appl 2012, 35 (2012). https://doi.org/10.1186/1029-242X-2012-35

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• DOI: https://doi.org/10.1186/1029-242X-2012-35

### Keywords

• analytic
• p-valent
• integral operator
• argument 