Some Starlikeness Conditions for the Analytic Functions and Integral Transforms

Let be the class of analytic functions of the form f(z)=z+\sum_{k=n+1}^{\infty} a_kz^k \qquad z\in\Delta, where n\in {\mathbb N} is fixed. For and we define a new class of non-Bazilevi? analytic functions by {\mathcal U}_n(\alpha,\lambda,\mu)=\left\{f\in {\mathcal A}_n:\left|(1-\alpha)\left(\frac{z}{f(z)}\right)^\mu+ \alpha fʹ(z)\left(\frac{z}{f(z)}\right)^{\mu+1}-1\right| < \lambda,\qquad z\in\Delta\right\}. In this paper we find conditions on \lambda and so that included in and Here and denote the usual classes of strongly starlike functions of order \gamma and starlike function of order respectively. For c+1-\mu > 0, we define an integral transform and find conditions on and c so that it will belong to a subclass of starlike functions. Keywords: Univalent; starlike and convex functions; Subordination and integral transform 1 Introduction and Preliminaries Let be the class of analytic functions defined on the unit disc with the normalized condition Let {\mathcal S} be the class of all functions which are univalent in \Delta. So has the form f(z)=z+\sum_{k=n+1}^{\infty} a_kz^k, \qquad z\in\Delta. (1.1) Here {\mathcal A}_1={\mathcal A}. A function is said to be in {\mathcal S}^* iff is starlike domain with respect to the origin. Let 0\leq\beta < 1 and {\mathcal S}^*(\beta)=\left\{f\in {\mathcal A}: {\rm Re}\frac{z fʹ(z)}{f(z)} > \beta, z\in\Delta\right\} be the class of starlike functions of order So A function is said to be the strongly starlike of order iff satisfies the condition \frac{zfʹ(z)}{f(z)}\prec\left(\frac{1+z}{1-z}\right)^\gamma, \qquad z\in\Delta, where \prec denotes the subordination (for basic results in subordination we refer to [3]). We denote to be the class of strongly starlike functions of order Clearly If then is completely contained in the class of bounded starlike functions [2]. Here and {\mathcal S}_\gamma^n \equiv {\mathcal S}_\gamma\bigcap{\mathcal A}_n. For and we define the class by {\mathcal U}_n(\alpha,\lambda,\mu)=\left\{f\in {\mathcal A}_n:\left|(1-\alpha)\left(\frac{z}{f(z)}\right)^\mu+ \alpha fʹ(z)\left(\frac{z}{f(z)}\right)^{\mu+1}-1\right| < \lambda,\qquad z\in\Delta\right\}. and this class becomes which was introduced by Ozaki and Nunokawa [8]. They shown that For and this class becomes which is a subclass of the class of Bazilevi? functions. It is known that the class of Bazilevi? functions are univalent [1] and hence But since Koebe function does not belong to for and for that reason many authors were interested for this class. For different choices of with this class has been extensively studied by many authors which are included in [4, 5, 6, 7, 9, 10, 11]. In [13], Zhu has considered the above class for and established some results using differential subordination. In the present paper we consider this class for and by using different techniques we find conditions on \lambda and so that this subclass is included in different well-known subclasses like and Also in this paper we studied a generalized integral transform for this class. Some of the results are improved versions of the results found in Zhu[13] and some of the results are new.