Some Inequalities on p-Valent Functions Related to Geometric Structure Based on q-Derivative

Shahram Najafzadeh¹, Deborah Olufunmilayo Makinde²

¹Department of Mathematics, Payame Noor University, Tehran, Iran.
²Department of Mathematics, Obafemi Awolowo University, Ile-Ife, Osun State, Nigeria.

DOI: 10.4236/jamp.2020.82024                

Abstract

By applying the q-derivative, we introduce two new subclasses of p-valent functions with positive coefficients. By means of the well-known Jack’s lemma, some inequalities related to starlike, convex and close-to-convex functions are also obtained.

Keywords

p-Valent Functions, Jack’s Lemma, Starlike, Convex and Close-to-Convex Functions

1. Introduction

By ${\mathcal{A}}_p\left(n\right)$, we denote the class of functions of the type:

$f\left(z\right)=z^p+{\displaystyle \underset{k=n+p}{\overset{+\infty }{\sum }}{a}_k{z}^k}\mathrm{,}\left(n\mathrm{,}p\in \mathbb{N}\right)\mathrm{,}$ (1)

which are p-valent and analytic in the open unit disk $\mathbb{D}=\left\{z\in ℂ:\left|z\right|<1\right\}$, see [1].

Now, we introduce some basic definitions and related details of the q-calculus, see [2] [3] [4].

The q-shifted factorial is defined for $\alpha \mathrm{,}q\in ℂ$ as a product of n factors by:

${\left(\alpha ;q\right)}_n=(\begin{array}{ll}1,\hfill & n=0,\hfill \\ \left(1-\alpha \right)\left(1-\alpha q\right)\cdots \left(1-\alpha q^{n-1}\right),\hfill & n\in \mathbb{N},\hfill \end{array}$ (2)

and according to the basic analogue of the gamma function, we get:

${\left(q^{\alpha };q\right)}_n=\frac{{\left(1-q\right)}^n{\Gamma }_q\left(\alpha +n\right)}{{\Gamma }_q\left(\alpha \right)},\left(n>0\right),$ (3)

where the q-gamma function is given by:

${\Gamma }_q\left(x\right)=\frac{{\left(q;q\right)}_{\infty }{\left(1-q\right)}^{1-x}}{{\left(q^x;q\right)}_{\infty }},\left(0<q<1\right).$ (4)

If $\left|q\right|<1$ the relation (2) is meaningful for $n=\infty$ as a convergent product defined by:

${\left(\alpha ;q\right)}_{\infty }={\displaystyle \underset{j=0}{\overset{\infty }{\prod }}\left(1-\alpha q^j\right)}\mathrm{.}$ (5)

Further, we conclude that

${\Gamma }_q\left(x+1\right)=\frac{\left(1-q^x\right){\Gamma }_q\left(x\right)}{1-q}.$ (6)

For $0<q<1$, the q-derivative of a function f is defined by:

${\partial }_{q}f\left(z\right)=\frac{f\left(qz\right)-f\left(z\right)}{z\left(q-1\right)},\left(z\ne 0,q\ne 1\right).$ (7)

A simple calculation yields that for $m\in \mathbb{N}$ and $\lambda >-1$,

${\partial }_q^m{z}^{\lambda }=\frac{{\Gamma }_q\left(\alpha \right)\left(1+\lambda \right)}{{\Gamma }_q\left(\alpha \right)\left(1+\lambda -m\right)}{z}^{\lambda -m}.$ (8)

Also, in view of the following relation:

$\underset{q\to 1^{-}}{\lim }\frac{{\left(q^{\alpha };q\right)}_n}{{\left(1-q\right)}^n}={\left(\alpha \right)}_n\mathrm{,}$ (9)

we note that the q-shifted factorial (2) reduces to the well-known Pochhammer symbol ${\left(\alpha \right)}_n$ [5], which is defined by:

${\left(\alpha \right)}_n=(\begin{array}{ll}1,\hfill & n=0,\hfill \\ \alpha \left(\alpha +1\right)\cdots \left(\alpha +n-1\right),\hfill & n\in \mathbb{N}.\hfill \end{array}$

Differentiating (1) m times with respect to z (8), we conclude

${\partial }_q^{m}f\left(z\right)=\frac{{\Gamma }_q\left(1+p\right)}{{\Gamma }_q\left(1+p-m\right)}{z}^{p-m}+{\displaystyle \underset{k=n+p}{\overset{\infty }{\sum }}\frac{{\Gamma }_q\left(1+k\right)}{{\Gamma }_q\left(1+k-m\right)}{a}_k{z}^{k-m}}.$ (10)

A function $f\left(z\right)\in {\mathcal{A}}_p\left(n\right)$ is said to be in the subclass $X_p\left(n\mathrm{,}m\right)$ if it satisfies the inequality:

$\left|\frac{{\Gamma }_q\left(1+p-m\right)}{{\Gamma }_q\left(1+p\right)}\frac{{\partial }_q^{m}f\left(z\right)}{z^{p-m}}-1\right|<1,$ (11)

where $z\in \mathbb{D}$, $p\in \mathbb{N}$, $0<q<1$ and $m\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$. Indeed $f\left(z\right)\in {\mathcal{A}}_p\left(n\right)$ is said to be in the subclass $Y_p\left(n\mathrm{,}m\right)$ if it satisfies the inequality:

$\left|\frac{z\left({\partial }_q^{m}f\left(z\right)\right)}{{\partial }_q^{m}f\left(z\right)}-\left(p-m\right)\right|<p-m.$ (12)

For details see [6].

2. Main Results

To prove the main theorems related to $X_p\left(n\mathrm{,}m\right)$ and $Y_p\left(n\mathrm{,}m\right)$, we need the following lemma due to Jack [7] [8].

Lemma 1. Let $w\left(z\right)$ e non-constant in $\mathbb{D}$ and $w\left(0\right)=0$. If $\left|w\right|$ attains its maximum value on the circle $\left|z\right|=r<1$ at $z_0$, then $z_0{w}^{\prime }\left(z_0\right)=tw\left(z_0\right)$, where $t\ge 1$ is a real number.

A function $f\left(z\right)\in {\mathcal{A}}_p\left(n\right)$ is said to be in the subclass ${\mathcal{A}}_p\mathcal{K}\left(n\right)$ of p-valently close-to-convex functions with respect to the origin in $\mathbb{D}$ if

$\mathrm{Re}\left\{\frac{f^{\prime }\left(z\right)}{z^{p-1}}\right\}>0,\left(z\in \mathbb{D},p\in \mathbb{N}\right).$

Also, $f\left(z\right)\in {\mathcal{A}}_p\mathcal{K}\left(n\right)$ is said to be in the subclass ${\mathcal{A}}_p\mathcal{S}\left(n\right)$ of p-valently starlike functions with respect to the origin in $\mathbb{D}$ if

$\mathrm{Re}\left\{\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right\}>0,\left(z\in \mathbb{D},p\in \mathbb{N}\right).$

Further $f\left(z\right)\in {\mathcal{A}}_p\left(n\right)$ is said to be in the subclass ${\mathcal{A}}_p\mathcal{C}\left(n\right)$ of p-valently convex functions with respect to the origin in $\mathbb{D}$ if

$\mathrm{Re}\left\{1+\frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}\right\}>0,$

see [9] [10].

Theorem 2. If $f\left(z\right)\in {\mathcal{A}}_p\left(n\right)$ satisfies the inequality:

$\left\{\frac{z\left({\partial }_q^{m}f\left(z\right)\right)}{{\partial }_q^{m}f\left(z\right)}-\left(p-m\right)\right\}<\frac{1}{2},$ (13)

then $f\left(z\right)\in X_p\left(n\mathrm{,}m\right)$.

Proof. Let $f\left(z\right)\in {\mathcal{A}}_p\left(n\right)$, we define the function $w\left(z\right)$ by:

$\frac{{\Gamma }_q\left(1+p-m\right)}{{\Gamma }_q\left(1+p\right)}\frac{{\partial }_q^{m}f\left(z\right)}{z^{p-m}}=1+w\left(z\right),\left(z\in \mathbb{D},p\in \mathbb{N},n\in {\mathbb{N}}_0\right).$ (14)

with a simple calculation we have $w\left(0\right)=0$ (in $\mathbb{U}$ ).

For (14), we obtain:

$\frac{{\Gamma }_q\left(1+p-m\right)}{{\Gamma }_q\left(1+p\right)}{\partial }_q^{m}f\left(z\right)=z^{p-m}+z^{p-m}w\left(z\right),$

or

$\frac{{\Gamma }_q\left(1+p-m\right)}{{\Gamma }_q\left(1+p\right)}{\left({\partial }_q^{m}f\left(z\right)\right)}^{\prime }=\left(p-m\right)z^{p-m-1}+\left(p-m\right)z^{p-m-1}w\left(z\right)+z^{p-m}{w}^{\prime }\left(z\right),$

or equivalently

$\frac{{\Gamma }_q\left(1+p-m\right)}{{\Gamma }_q\left(1+p\right)}\frac{{\left({\partial }_q^{m}f\left(z\right)\right)}^{\prime }}{z^{p-m-1}}=\left(p-m\right)\left(1+w\left(z\right)\right)+z{w}^{\prime }\left(z\right).$ (15)

From (14) and (15), we get:

$\frac{z{w}^{\prime }\left(z\right)}{1+w\left(z\right)}=\frac{z{\left({\partial }_q^{m}f\left(z\right)\right)}^{\prime }}{{\partial }_q^{m}f\left(z\right)}-\left(p-m\right)\mathrm{.}$ (16)

Now, let for $z_0\in �$, $\underset{\left|z\right|\le \left|z_0\right|}{max}\left|w\left(z\right)\right|=\left|w\left(z_0\right)\right|=1$, then by using Jack’s lemma and putting $w\left(z_0\right)={\text{e}}^{i\theta }\ne -1$ in (16), we have:

$\begin{array}{l}\mathrm{Re}\left\{\frac{z{\left({\partial }_q^{m}f\left(z\right)\right)}^{\prime }}{{\partial }_q^{m}f\left(z\right)}-\left(p-m\right)\right\}=\left\{\frac{z_0{w}^{\prime }\left(z_0\right)}{1+w\left(z_0\right)}\right\}=\mathrm{Re}\left\{\frac{tw\left(z_0\right)}{1+w\left(z_0\right)}\right\}\\ =\mathrm{Re}\left\{\frac{t{\text{e}}^{i\theta }}{1+{\text{e}}^{i\theta }}\right\}=\mathrm{Re}\left\{\frac{t\left(\cos \theta +i\sin \theta \right)}{\left(1+\cos \theta \right)+i\sin \theta }\right\}\\ =\mathrm{Re}\left\{\frac{t\left(\cos \theta +i\sin \theta \right)\left(\left(1+\cos \theta \right)-i\sin \theta \right)}{\left(1+\cos \theta \right)+i\sin \theta \left(\left(1+\cos \theta \right)-i\sin \theta \right)}\right\}\\ =\mathrm{Re}\left\{\frac{t\left(1+\cos \theta +i\sin \theta \right)}{2+2\cos \theta }\right\}\\ =\mathrm{Re}\left\{\frac{t\left(1+\cos \theta \right)}{2+2\cos \theta }+\frac{it\sin \theta }{2+2\cos \theta }\right\}=\frac{t}{2}\ge \frac{1}{2},\end{array}$

which is a contradiction with (13). Thus we have $\left|w\left(z\right)\right|<1$ for all $z\in \mathbb{D}$, so from (14) we conclude:

$\left|\frac{{\Gamma }_q\left(1+p-m\right)}{{\Gamma }_q\left(1+p\right)}\frac{{\partial }_q^{m}f\left(z\right)}{z^{p-m}}-1\right|=\left|w\left(z\right)\right|<1,$

and this gives the result.

By letting $m=0$ and ( $m=1,q\to 1$ ), we have the following corollaries which are due to Irmak and Cetin [11].

Corollary 3. If $f\left(z\right)\in {\mathcal{A}}_p\left(n\right)$ satisfies

$\mathrm{Re}\left\{\frac{z{f}^{\prime }}{f}-p\right\}<\frac{1}{2},\left(z\in \mathbb{D},p\in \mathbb{N}\right),$

then $\left|\frac{f\left(z\right)}{z^p}-1\right|<1$.

Corollary 4. If $f\left(z\right)\in {\mathcal{A}}_p\left(n\right)$ satisfies the inequality

$\mathrm{Re}\left\{1+\frac{z{f}^{″}}{f^{\prime }}-p\right\}<\frac{1}{2},\left(z\in \mathbb{D},p\in \mathbb{N}\right),$

then $f\left(z\right)\in {\mathcal{A}}_p\mathcal{K}\left(n\right)$ and $\left|\frac{f^{\prime }}{z^{p-1}}-p\right|<p$.

Theorem 5. If $f\left(z\right)\in {\mathcal{A}}_p\left(n\right)$ satisfies

$\left\{1+\left[\frac{{\left({\partial }_q^{m}f\left(z\right)\right)}^{\prime \text{}\prime }}{{\left({\partial }_q^{m}f\left(z\right)\right)}^{\prime }}-\frac{{\left({\partial }_q^{m}f\left(z\right)\right)}^{\prime }}{{\partial }_q^{m}f\left(z\right)}\right]\right\}<\frac{1}{2},\left(z\in \mathbb{D},p\in \mathbb{N},n\in {\mathbb{N}}_0\right),$ (17)

then $f\left(z\right)\in Y_p\left(n\mathrm{,}m\right)$.

Proof. Let the function $f\left(z\right)\in {\mathcal{A}}_p\left(n\right)$, we define the function $w\left(z\right)$ by

$\frac{z{\left({\partial }_q^{m}f\left(z\right)\right)}^{\prime }}{{\partial }_q^{m}f\left(z\right)}=p\left(1+w\left(z\right)\right)\mathrm{.}$ (18)

It is easy to verify that $w\left(z\right)$ is analytic in $\mathbb{D}$ and $w\left(0\right)=0$. By (18), we have:

$z{\left({\partial }_q^{m}f\left(z\right)\right)}^{\prime }=p{\partial }_q^{m}f\left(z\right)+p{\partial }_q^{m}f\left(z\right)w\left(z\right),$

or

$\begin{array}{l}{\left({\partial }_q^{m}f\left(z\right)\right)}^{\prime }+z{\left({\partial }_q^{m}f\left(z\right)\right)}^{\prime \text{}\prime }\\ =p{\left({\partial }_q^{m}f\left(z\right)\right)}^{\prime }+p\left(w^{\prime }\left(z\right){\partial }_q^{m}f\left(z\right)+w\left(z\right){\left({\partial }_q^{m}f\left(z\right)\right)}^{\prime }\right),\end{array}$

or

$1+\frac{z{\left({\partial }_q^{m}f\left(z\right)\right)}^{\prime \text{}\prime }}{{\left({\partial }_q^{m}f\left(z\right)\right)}^{\prime }}=p\left(1+w\left(z\right)\right)+p{w}^{\prime }\left(z\right)\frac{{\partial }_q^{m}f\left(z\right)}{{\left({\partial }_q^{m}f\left(z\right)\right)}^{\prime }}\mathrm{,}$

or by (18) we get

$1+\frac{z{\left({\partial }_q^{m}f\left(z\right)\right)}^{\prime \text{}\prime }}{{\left({\partial }_q^{m}f\left(z\right)\right)}^{\prime }}=p\left(1+w\left(z\right)\right)+\frac{z{w}^{\prime }\left(z\right)}{1+w\left(z\right)}\mathrm{.}$

Now, let for a point $z_0\in \mathbb{D}$, $\underset{\left|z\right|\le \left|z_0\right|}{\max }\left|w\left(z\right)\right|=\left|w\left(z_0\right)\right|=1$. By Jack’s lemma and putting $w\left(z_0\right)={\text{e}}^{i\theta }$ we conclude:

$\begin{array}{l}\mathrm{Re}\left\{1+z\left[\frac{{\left({\partial }_q^{m}f\left(z\right)\right)}^{\prime \text{}\prime }}{{\left({\partial }_q^{m}f\left(z\right)\right)}^{\prime }}-\frac{{\left({\partial }_q^{m}f\left(z\right)\right)}^{\prime }}{{\partial }_q^{m}f\left(z\right)}\right]\right\}\\ =\mathrm{Re}\left\{\frac{z_0{w}^{\prime }\left(z_0\right)}{1+w\left(z_0\right)}\right\}=\mathrm{Re}\left\{\frac{tw\left(z_0\right)}{1+w\left(z_0\right)}\right\}=\mathrm{Re}\left\{\frac{t{\text{e}}^{i\theta }}{1+{\text{e}}^{i\theta }}\right\}>\frac{t}{2}\ge \frac{1}{2},\end{array}$

which is contradiction with (17). Thus for all $z\in \mathbb{D}$, $\left|w\left(z\right)\right|<1$ and so from (18), we have:

$\left|\frac{z{\left({\partial }_q^{m}f\left(z\right)\right)}^{\prime }}{{\partial }_q^{m}f\left(z\right)}-p\right|<p,$

thus the proof is complete.

By letting $m=0$ and ( $m=1,q\to 1$ ) we have the following corollaries that the first one is due to Irmak and Cetin [5].

Corollary 6. If $f\left(z\right)\in {\mathcal{A}}_p\left(n\right)$ satisfies the inequality

$\mathrm{Re}\left\{1+z\left(\frac{f^{″}}{f^{\prime }}-\frac{f^{\prime }}{f}\right)\right\}<\frac{1}{2},\left(z\in \mathbb{D},p\in \mathbb{N}\right),$

then $f\left(z\right)\in {\mathcal{A}}_p\mathcal{S}\left(n\right)$ and $\left|\frac{z{f}^{\prime }}{f}-p\right|<p$.

3. Conclusion

Studying the theory of analytic functions has been an area of concern for many authors. Literature review indicates lots of researches on the classes of p-valent analytic functions. The interplay of geometric structures is a very important aspect in complex analysis. In this study, two new subclasses of p-valent functions were defined by using q-analogue of the well-known operators and we gave some geometric structures like starlike, convex and close-to-convex properties of the subclasses. It is noted that the study is an extension of some previous studies as it is shown in corollaries 3, 4, 6.

Acknowledgements

The authors wish to thank the reviewer for their valuable suggestions which add to the quality of this paper.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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