First-Order Differential Subordinations and Their Applications

We let $\mathcal{H}$ denote the class of analytic functions in the open unit disk $\mathbb{U}=\{z\in \mathbb{C}:|z|<1\}$ and define where $a\in \mathbb{C}$ and n is a positive integer number. Furthermore, we introduce the subclass ${\mathcal{A}}_n$ of $\mathcal{H}$ as follows:

In particular, we set ${\mathcal{A}}_1\equiv \mathcal{A}$. As usual, let the subclass $𝒮$ of $\mathcal{A}$ be the class of all univalent functions in the open unit disk $\mathbb{U}$. A function $f\in$$\mathcal{A}$ is said to be starlike of order $\gamma (0\le \gamma <1)$, written $f\in {𝒮}^{\ast }\left(\gamma \right)$, if it satisfies

Specifically, we put ${𝒮}^{\ast }\left(0\right)\equiv {𝒮}^{\ast }$. Every element in ${𝒮}^{\ast }$ is called a starlike function. Furthermore, a function $f\in \mathcal{A}$ is said to be convex of order $\gamma (0\le \gamma <1),$ written $f\in \mathcal{K}\left(\gamma \right)$, if it satisfies

In particular, we put $\mathcal{K}\left(0\right)\equiv \mathcal{K}$. Every element in $\mathcal{K}$ is called a convex function. Now for analytic functions in $\mathbb{U}$ with the fixed initial coefficient, we define the class ${\mathcal{H}}_{\beta }[a,n]$ as follows: where n is a positive integer number, $a\in \mathbb{C}$ and $\beta \in \mathbb{C}$ are fixed numbers. Moreover we assume where n is a positive integer number and $b\in \mathbb{C}$ is a fixed number. In addition, we set ${\mathcal{A}}_{1,b}\equiv {\mathcal{A}}_b$. Assume f and g be in $\mathcal{H}$. We say that the function f is subordinate to g, denoted by $f\prec g$, if there exists an analytic function $\omega$ in $\mathbb{U}$, with $\omega \left(0\right)=0$ and $\left|\omega \right(z\left)\right|\le \left|z\right|<1$, such that $f\left(z\right)=g\left(\omega \right(z\left)\right)$. Moreover, if g is an univalent function in $\mathbb{U}$, then $f\prec g$ if and only if $f\left(0\right)=0$ and $f\left(\mathbb{U}\right)\subset g\left(\mathbb{U}\right)$.

By considering the function $\frac{1+Az}{1+Bz}$, we can generalize the class of starlike functions as follows:

Let $f\in \mathcal{A}$ and $-1\le B<A\le 1$. Then it is said that $f\in {𝒮}^{\ast }[A,B]$ if and only if

Through (1), it can be easily observed that $\frac{1+Az}{1+Bz}$ with $-1<B<A<1$ maps the open unit disc $\mathbb{U}$ onto the open disc with the center $C=(1-AB)/(1-B^2)$ and the radius $R=(A-B)/(1-B^2)$. Thus, for all $f\in {𝒮}^{\ast }[A,B]$, the relation $\frac{1-A}{1-B}<\mathfrak{Re}\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}<\frac{1+A}{1+B}$ holds and hence ${𝒮}^{\ast }[A,B]\subset {𝒮}^{\ast }\left(\frac{1-A}{1-B}\right)$. Moreover for the different values of A and B, other types of the class $\mathcal{A}$ such as the class ${𝒮}^{\ast }[1,-1]$, which is equivalent to the class ${𝒮}^{\ast }$, also for $0\le \alpha <1$, the class ${𝒮}^{\ast }[1-2\alpha ,-1]$ which is equivalent to $S^{\ast }\left(\alpha \right)$ and ${𝒮}^{\ast }[\alpha ,0]=\left\{f\in \mathcal{A}:\left|\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}-1\right|<\alpha \right\}$ are obtained.

The theory of differential subordination has a key role in the study of geometric function theory. In 1935 Goluzin [1] considered the subordination $z{p}^{\prime }\left(z\right)\prec h\left(z\right)$ and proved that if h is convex then $p\left(z\right)\prec q\left(z\right)={\int }_0^{z}h\left(t\right)t^{-1}dt$. Furthermore, in 1970 Suffridge [2] showed that Goluzin’s result is true if h is starlike. Moreover, Miller and Mocanu by writing many research papers in this direction extended the concept of differential subordination (see for example [3] and references therein). Further, many authors have recently using different combinations of the representations of starlike and convex functions have obtained the simple conditions for starlikeness and convexity of analytic functions. In [4], Silverman gained the results for analytic functions including the terms as the quotient of the analytic representations of convex and starlike functions. For instance, Silverman proved that $G_b\subset {𝒮}^{\ast }\left(\frac{2}{1+\sqrt{1+8b}}\right)$, where $0<b\le 1$ and

Next, Obradovic and Tuneski [5], in view of $\frac{1}{1+b}\ge \frac{2}{1+\sqrt{1+8b}}$, improved the work of Silverman. Indeed, they established $G_b\subset {𝒮}^{\ast }\left(\frac{1}{1+b}\right)\subset {𝒮}^{\ast }\left(\frac{2}{1+\sqrt{1+8b}}\right)$ for $0<b\le 1$. Nunokawa et al. [6], by applying the Silverman’s quotient function [4], proved that f could be strongly starlike, strongly convex or starlike in $\mathbb{U}$. In [7,8,9,10], the authors have studied some conditions for the analytic functions to belong to the class ${𝒮}^{\ast }[A,B]$. In [11], some results related to the above discussion, with respect to n-symmetric points of functions, have been given. Inspired by [9,10], in this paper, we will extend and improve some results obtained in [9,10] and then we will determine some conditions which by means of them, a function belongs to the class ${𝒮}^{\ast }[A,B]$.

The contents of this article are regulated as follows: In Section 3, initially, we will prove a theorem that is the extension of a little change to the Suffridge theorem [3]. Next, we bring some applications of this theorem as the main results, making the functions be in the class ${𝒮}^{\ast }[A,B]$. These results extend and improve some results in [10]. In Section 4, we intend to bring some sufficient conditions for starlikeness of analytic functions. We also produce the functions belonging to the class ${𝒮}^{\ast }[A,B]$ by considering other conditions, and so we include some corollaries from the result acquired. Furthermore, these results extend and improve some results in [9]. Moreover, Suffridge’s result is used in recent investigations like [12,13,14,15]. Note that some results related to this article for analytic functions with fixed initial coefficients are also mentioned.

In the continuation of the argument, in order to prove the main results, we require to remind a definition and two basic lemmas:

First, we mention a lemma which is slightly different from the original one, ([3], Th 3.4 h).

By considering $\theta \left(z\right)\equiv 0$ in Lemma 3, we extend a little change of the Suffridge theorem [3] as follows:

Using the same argument of Lemma 3 and applying Lemma 1, we obtain the following theorem, and we omit its proof:

By putting $\varphi \equiv 1$ in Lemma 3 and Theorem 1, we reach to the following corollaries:

By applying the same argument as Corollary 4, and using Corollary 3 we have:

Setting $p\left(z\right)=\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}$ in Corollary 4, we obtain:

Setting $p\left(z\right)=\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}$ in Corollary 5, we obtain:

Putting $A=1$ in Corollary 6, we get to the following corollary:

Putting $A=1$ in Corollary 7, we come to the following corollary:

It is well known that for $f\in \mathcal{A}$, the condition $|z{f}^{\prime \prime }\left(z\right)/f^{\prime }\left(z\right)|<2$ is sufficient for starlikeness of f. In this section, we will extend this result and will also try to bring other sufficient conditions for starlikeness.

By putting $A=1,B=-1$ and $p\left(z\right)=\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}$ in Theorem 2 we obtain:

By using Corollary 10, we have:

By putting $A=0$, $B=-1$ and $p\left(z\right)=\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}$ in Theorem 2, we obtain:

We remark that Corollary 12 is the generalization of Marx-Strohh$\ddot{a}$cker Theorem (see [17]). By putting $A=0$, $0<b=-B<1$ and $p\left(z\right)=\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}$ in Theorem 2 we gain:

With the same approach as the previous theorem and by applying Lemma 1, we attain the following theorem which we omit the proof of.

By putting $A=0$ and letting $N(0,B,n)=(n+1)\frac{1-\alpha }{\alpha }$, we obtain $B=\frac{\alpha -1}{\alpha }$, where $N(A,B,n)$ is given in Theorem 4. Now let $\alpha >0$. Since $-1\le B<A\le 0$, we have $\frac{1}{2}\le \alpha <1$. Hence we gain:

By setting $A=0$, $b=\frac{-B}{n}$ and letting $N(0,B,n,\frac{B}{n})=(n+1)\frac{1-\alpha }{\alpha }$, we obtain $B=\frac{\alpha -1}{\alpha }$, where $N(A,B,n,b)$ is given in Theorem 4. Now let $\alpha >0$. Since $-1\le B<A\le 0$, we have $\frac{1}{2}\le \alpha <1$. So we obtain:

Therefore making use of Corollary 16, if $\frac{1}{2}\le \alpha <1$ and $\left|\lambda \right|\le arctan\left(\sqrt{\frac{3(1-\alpha )}{\alpha }}\right)$, then $f\in S^{\ast }\left(\alpha \right)$. In addition, if $\frac{4}{5}\le \alpha <1$ and $\left|\lambda \right|\le arctan\left(\sqrt{\frac{3(1-\alpha )}{\alpha }}\right)$, then f is convex.

Finally we prove the following result:

In this paper, we have proved a result similar to Suffridge’s theorem and given some applications related to this result. Moreover, we have investigated some sufficient conditions for starlikeness of analytic functions and functions that are in the class ${𝒮}^{\ast }[A,B]$.