Some properties of an integral operator defined by convolution

Abstract

In this investigation, motivated from Breaz study, we introduce a new family of integral operator using famous convolution technique. We also apply this newly defined operator for investigating some interesting mapping properties of certain subclasses of analytic and univalent functions.

2010 Mathematics Subject Classification: 30C45; 30C10.

1. Introduction

Let A denote the class of analytic function satisfying the condition f(0) = f'(0) - 1 = 0 in the open unit disc $\mathbb{U}=\left\{z:\mid z\mid <1\right\}.$ By S, C, S*, C*, and K we means the well-known subclasses of A which consist of univalent, convex, starlike, quasi-convex, and close-to-convex functions, respectively. The well-known Alexander-type relation holds between the classes C and S* and C* and K, that is,

$f\left(z\right)\in C⇔z{f}^{\prime }\left(z\right)\in {S}^{*},$

and

$f\left(z\right)\in {C}^{*}⇔z{f}^{\prime }\left(z\right)\in K.$

It was proved in [1] that a locally univalent function f(z) is close-to-convex, if and only if

$\underset{{\theta }_{1}}{\overset{{\theta }_{2}}{\int }}\mathsf{\text{Re}}\phantom{\rule{1em}{0ex}}\left\{1+\frac{z{f}^{″}\left(z\right)}{{f}^{\prime }\left(z\right)}\right\}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}d\theta >-\pi ,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}z=r{e}^{i\phantom{\rule{0.3em}{0ex}}\theta },$
(1.1)

for each r (0,1) and every pair θ1, θ2 with 0 ≤ θ1< θ2 ≤ 2π.

Let P k (ξ) be the class of functions p(z) analytic in $\mathbb{U}$ with p(0) = 1 and

$\underset{0}{\overset{2\pi }{\int }}\left|\frac{\mathsf{\text{Re}}\phantom{\rule{1em}{0ex}}p\left(z\right)-\xi }{1-\xi }\right|\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}d\theta \le k\pi ,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}z=r{e}^{i\phantom{\rule{0.3em}{0ex}}\theta },\phantom{\rule{2.77695pt}{0ex}}k\ge 2.$

This class was introduced in [2] and for k = 2, ξ = 0, the class p k (ξ) reduces to the class P of functions with positive real part. We consider the following classes:

$\begin{array}{ccc}{R}_{k}\left(\xi \right)& =& \left\{f\left(z\right)\in A:\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\in {P}_{k}\left(\xi \right),\phantom{\rule{2.77695pt}{0ex}}z\in \mathbb{U}\right\}\hfill \\ {T}_{k}\left(\xi \right)& =& \left\{f\left(z\right)\in A:\exists \phantom{\rule{2.77695pt}{0ex}}g\left(z\right)\in C:\frac{{f}^{\prime }\left(z\right)}{{g}^{\prime }\left(z\right)}\in {P}_{k}\left(\xi \right),\phantom{\rule{2.77695pt}{0ex}}z\in \mathbb{U}\right\}.\end{array}$

These classes were studied by Noor [35] and Padmanabhan and Parvatham [2]. Also it can easily be seen that R2(0) = S* and T2(0) = K, where S* and K are the well-known classes of starlike and close-to-convex functions.

Using the same method as that of Kaplan [1], Noor [6] extend the result of Kaplan given in (1.1), and proved that a locally univalent function f(z) is in the class T k , if and only if

$\underset{{\theta }_{1}}{\overset{{\theta }_{2}}{\int }}\mathsf{\text{Re}}\phantom{\rule{1em}{0ex}}\left\{1+\frac{z{f}^{″}\left(z\right)}{{f}^{\prime }\left(z\right)}\right\}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}d\theta >-\frac{k}{2}\pi ,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}z=r{e}^{i\phantom{\rule{0.3em}{0ex}}\theta },$
(1.2)

for each r (0,1) and every pair θ1, θ2 with 0 ≤ θ1 < θ2 ≤ 2π

For any two analytic functions

the convolution (Hadamard product) of f(z) and g(z) is defined by

$f\left(z\right)*\mathsf{\text{g}}\left(z\right)=\sum _{n=0}^{\infty }{a}_{n}\phantom{\rule{0.3em}{0ex}}{b}_{n}\phantom{\rule{0.3em}{0ex}}{z}^{n},\phantom{\rule{0.3em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\left(z\in \mathbb{U}\right).$

Using the techniques from convolution theory many authors generalized Breaz operator in several directions, see [7, 8] for example. Here, we introduce a generalized integral operator I n (f i , g i , h i )(z): AnA as follows

${I}_{n}\left({f}_{i},\phantom{\rule{0.3em}{0ex}}{g}_{i},\phantom{\rule{0.3em}{0ex}}{h}_{i}\right)\left(z\right)=\underset{0}{\overset{z}{\int }}{\prod _{i=1}^{n}\left({\left({f}_{i}\left(t\right)\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}*\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{g}_{i}\left(t\right)\right)}^{\prime }\right)}^{{\alpha }_{i}}{\left(\frac{{h}_{i}\left(t\right)}{t}\right)}^{{\beta }_{i}}dt,$
(1.3)

where f i (z), g i (z), h i (z) A with f i (z) * g i (z) ≠ 0 and α i , β i ≥ 0 for i = 1, 2,..., n. The operator I n (f i , g i , h i )(z) reduces to many well-known integral operators by varying the parameters α i , β i and by choosing suitable functions instead of f i (z), g i (z). For example,

1. (i)

If we take we obtain the integral operator

${I}_{n}\left({f}_{i},\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{h}_{i}\right)\left(z\right)=\underset{0}{\overset{z}{\int }}{\prod _{i=1}^{n}\left({{f}_{i}}^{\prime }\left(t\right)\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\right)}^{{\alpha }_{i}}{\left(\frac{{h}_{i}\left(t\right)}{t}\right)}^{{\beta }_{i}}dt,$
(1.4)

introduced in [9].

1. (ii)

If we take α i = 0 and 1 ≤ in, we obtain the integral

${I}_{n}\left(\phantom{\rule{0.3em}{0ex}}{h}_{i}\right)\left(z\right)=\underset{0}{\overset{z}{\int }}\prod _{i=1}^{n}{\left(\frac{{h}_{i}\left(t\right)}{t}\right)}^{{\beta }_{i}}dt,$

introduced and studied by Breaz and Breaz [10].

1. (iii)

If we take we obtain the integral operator

${I}_{n}\left({f}_{i}\right)\left(z\right)=\underset{0}{\overset{z}{\int }}{\prod _{i=1}^{n}\left({{f}_{i}}^{\prime }\left(t\right)\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\right)}^{{\alpha }_{i}}dt,$

introduced and studied by Breaz et al. [11].

1. (iv)

If we take n = 1, α 1 = 0 and β 1 = 1 in (1.4), we obtain the Alexander integral operator

${I}_{n}\left(\phantom{\rule{0.3em}{0ex}}{h}_{1}\right)\left(z\right)=\underset{0}{\overset{z}{\int }}\left(\frac{{h}_{1}\left(t\right)}{t}\right)dt,$

introduced in [12].

1. (v)

If we take n = 1, α 1 = 0 and β 1 = β, we obtain the integral operator

${I}_{n}\left(\phantom{\rule{0.3em}{0ex}}{h}_{1}\right)\left(z\right)=\underset{0}{\overset{z}{\int }}{\left(\frac{{h}_{1}\left(t\right)}{t}\right)}^{\beta }dt,$

studied in [13].

In this article, we study the mapping properties of different subclasses of analytic and univalent functions under the integral operator given in (1.3). To prove our main results, we need the following lemmas.

Lemma 1.1[14]. Let f(z) R k (ξ) for k ≤ 2, 0 ≤ ξ < 1. Then with 0 ≤ θ1 < θ2 ≤ 2π and z = re, r < 1,

$\underset{{\theta }_{1}}{\overset{{\theta }_{2}}{\int }}\mathsf{\text{Re}}\phantom{\rule{1em}{0ex}}\left\{\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right\}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}d\theta >-\left(\frac{k}{2}-1\right)\left(1-\xi \right)\pi .$

Lemma 1.2[15]. If f(z) C and g(z) K, then f(z)*g(z) K.

2. Main results

Theorem 2.1. Let f i (z) S*, g i (z) C* and h i (z) R k (ξ) with 0 ≤ ξ < 1, k ≥ 2 for all 1 ≤ in If

$\sum _{i=1}^{n}\left({\alpha }_{i}+\left(\frac{k}{2}-1\right)\left(1-\rho \right){\beta }_{i}\right)\le 1,$
(2.1)

then integral operator defined by (1.3) belongs to the class of close-to-convex functions.

Proof. Let f i (z) S* and g i (z) C*. Then there exists φ i (z) C such that

${f}_{i}\left(z\right)=z{{\phi }_{i}}^{\prime }\left(z\right).$

Now consider

${f}_{i}\left(z\right)*{g}_{i}\left(z\right)=z{{\phi }_{i}}^{\prime }\left(z\right)*{g}_{i}\left(z\right)={\phi }_{i}\left(z\right)*z{{g}_{i}}^{\prime }\left(z\right).$

Since g i (z) C*, then by Alexander-type relation $z{g}_{i}^{\prime }\left(z\right)\in K.$ So, by Lemma 1.2, we have

${\phi }_{i}\left(z\right)*z{{g}_{i}}^{\prime }\left(z\right)\in K,$

which implies that

${f}_{i}\left(z\right)*{g}_{i}\left(z\right)\in K$

and hence, by using (1.1),

$\underset{{\theta }_{1}}{\overset{{\theta }_{2}}{\int }}\mathsf{\text{Re}}\phantom{\rule{1em}{0ex}}\left\{1+\frac{z{\left({f}_{i}\left(z\right)*{g}_{i}\left(z\right)\right)}^{\prime \prime }}{{\left({f}_{i}\left(z\right)*{g}_{i}\left(z\right)\right)}^{\prime }}\right\}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}d\theta >-\pi .$
(2.2)

From (1.3), we obtain

${I}_{n}{\left({f}_{i},\phantom{\rule{0.3em}{0ex}}{g}_{i},\phantom{\rule{0.3em}{0ex}}{h}_{i}\right)}^{\prime }\left(z\right)=\prod _{i=1}^{n}{\left({\left({f}_{i}\left(z\right)\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}*\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{g}_{i}\left(z\right)\right)}^{\prime }\right)}^{{\alpha }_{i}}{\left(\frac{{h}_{i}\left(z\right)}{z}\right)}^{{\beta }_{i}}\mathsf{\text{.}}$
(2.3)

Differentiating (2.3) logarithmically, we have

Taking real part and then integrating with respect to θ, we get

Using (2.2) and Lemma 1.1, we have

$\begin{array}{c}\underset{{\theta }_{1}}{\overset{{\theta }_{2}}{\int }}\mathsf{\text{Re}}\phantom{\rule{1em}{0ex}}\left\{1+\frac{{I}_{n}{\left({f}_{i},\phantom{\rule{0.3em}{0ex}}{g}_{i},\phantom{\rule{0.3em}{0ex}}{h}_{i}\right)}^{\prime \prime }\left(z\right)}{{I}_{n}{\left({f}_{i},\phantom{\rule{0.3em}{0ex}}{g}_{i},\phantom{\rule{0.3em}{0ex}}{h}_{i}\right)}^{\prime }\left(z\right)}\right\}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}d\theta >-\pi \sum _{i=1}^{n}\left({\alpha }_{i}+\left(\frac{k}{2}-1\right)\left(1-\rho \right){\beta }_{i}\right)\\ \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}+\left(1-\sum _{i=1}^{\mathsf{\text{n}}}\left({\alpha }_{i}+{\beta }_{i}\right)\right)\left({\theta }_{\mathsf{\text{2}}}-{\theta }_{1}\right)\end{array}$

From (2.1), we can easily write

$\sum _{i=1}^{n}\left({\alpha }_{i}+{\beta }_{i}\right)<\sum _{i=1}^{n}\phantom{\rule{2.77695pt}{0ex}}\left({\alpha }_{i}+\left(\frac{k}{2}-1\right)\left(1-\rho \right){\beta }_{i}\right)\phantom{\rule{2.77695pt}{0ex}}\le 1\phantom{\rule{2.77695pt}{0ex}}.$

This implies that

$\sum _{i=1}^{n}\left({\alpha }_{i}+{\beta }_{i}\right)<1\phantom{\rule{2.77695pt}{0ex}},$

so, minimum is for θ1 = θ2, we obtain

$\underset{{\theta }_{1}}{\overset{{\theta }_{2}}{\int }}\mathsf{\text{Re}}\phantom{\rule{1em}{0ex}}\left\{1+\frac{{I}_{n}{\left({f}_{i},\phantom{\rule{0.3em}{0ex}}{g}_{i},\phantom{\rule{0.3em}{0ex}}{h}_{i}\right)}^{\prime \prime }\left(z\right)}{{I}_{n}{\left({f}_{i},\phantom{\rule{0.3em}{0ex}}{g}_{i},\phantom{\rule{0.3em}{0ex}}{h}_{i}\right)}^{\prime }\left(z\right)}\right\}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}d\theta >-\pi ,$

and this implies that I n (f i , g i , h i )(z) K.

For k = 2 in Theorem 2.1, we obtain

Corollary 2.3. Let f i (z) S*, g i (z) C* and h i (z) S*(ξ) with 0 ≤ ξ < 1, for all 1 ≤ i ≤ n. If

$\sum _{i=1}^{n}{\alpha }_{i}\le 1,$

then I n (f i , g i , h i )(z) K.

Theorem 2.4. Let f i (z) T k and h i (z) R k for 1 ≤ in. If α i , β i ≥ 0 such that α i + β i ≠ 0 and

$\sum _{i=1}^{n}\left(\frac{k}{2}\left({\alpha }_{i}+{\beta }_{i}\right)-{\beta }_{i}\right)\le 1,$
(2.4)

then I n (f i , h i )(z) defined by (1.4) belongs to the class of close-to-convex functions.

Proof. From (1.4), we have

${I}_{n}{\left({f}_{i},\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{h}_{i}\right)}^{\prime }\left(z\right)=\prod _{i=1}^{n}{\left({{f}_{i}}^{\prime }\left(z\right)\phantom{\rule{0.3em}{0ex}}\right)}^{{\alpha }_{i}}{\left(\frac{{h}_{i}\left(z\right)}{z}\right)}^{{\beta }_{i}}\mathsf{\text{.}}$
(2.5)

Differentiating (2.5) logarithmically, we have

Taking real part and then integrating with respect to θ, we get

where we have used Lemma 1.1 and (1.2)

$=-\phantom{\rule{2.77695pt}{0ex}}\sum _{i=1}^{n}\left(\left(\frac{k}{2}\right)\left({\alpha }_{i}+{\beta }_{i}\right)-{\beta }_{i}\right)+\left(1-\sum _{i=1}^{\mathsf{\text{n}}}\left({\alpha }_{i}+{\beta }_{i}\right)\right)\left({\theta }_{\mathsf{\text{2}}}-{\theta }_{1}\right)\phantom{\rule{0.3em}{0ex}}.$

From (2.4), we can obtain

$\sum _{i=1}^{n}\left({\alpha }_{i}+{\beta }_{i}\right)<1\phantom{\rule{2.77695pt}{0ex}}.$

So minimum is for θ1 = θ2, thus we have

$\underset{{\theta }_{1}}{\overset{{\theta }_{2}}{\int }}\mathsf{\text{Re}}\phantom{\rule{1em}{0ex}}\left(1+\frac{{I}_{n}{\left({f}_{i},\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{h}_{i}\right)}^{\prime \prime }\left(z\right)}{{I}_{n}{\left({f}_{i},\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{h}_{i}\right)}^{\prime }\left(z\right)}\right)\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}d\theta >-\pi \phantom{\rule{0.3em}{0ex}}.$

This implies that In(f i , h i )(z) K.

For k = 2 in Theorem 2.4, we obtain the following result.

Corollary 2.5. Let f i (z) K, h i (z) S* for 1 ≤ in and

$\sum _{i=1}^{n}\left(\frac{k}{2}\left({\alpha }_{i}+{\beta }_{i}\right)-{\beta }_{i}\right)\le 1\phantom{\rule{0.3em}{0ex}},$

then I n (f i , h i )(z) defined by (1.4) belongs to the class of close-to-convex functions.

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Acknowledgements

The authors would like to thank the reviewers and editor for improving the presentation of this article, and they also thank Dr. Ihsan Ali, Vice Chancellor AWKUM, for providing excellent research facilities in AWKUM.

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Correspondence to Muhammad Arif.

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The authors declare that they have no competing interests.

Authors' contributions

MA completed the main part of this article, KIN presented the ideas of this article, FG participated in some results of this article. MA made the text file and all the communications regarding the manuscript. All authors read and approved the final manuscript.

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Arif, M., Noor, K.I. & Ghani, F. Some properties of an integral operator defined by convolution. J Inequal Appl 2012, 13 (2012). https://doi.org/10.1186/1029-242X-2012-13