A certain class of starlike functions

AbstractThis paper presents a new class of functions analytic in the open unit disc, and closely related to the class of starlike functions. Besides being an introduction to this field, it provides an interesting connections defined class with well known classes. The paper deals with several ideas and techniques used in geometric function theory. The order of starlikeness in the class of convex functions of negative order is also considered here

A note on a class of pp-valent starlike functions of order beta

In this paper we obtain sharp coefficient bounds for certain $p$-valent starlike functions of order $\beta$, $0\le \beta<1$. Initially this problem was handled by Aouf in "M. K. Aouf, On a class of $p$-valent starlike functions of order $\alpha$, Internat. J. Math. $\&$ Math. Sci. 1987;10:733--744". We pointed out that the proof given by Aouf was incorrect and a correct proof is presented in this paper.Comment: 6 pages, 1 table, submitted to a journa

Subordination and Radius Problems for Certain Starlike Functions

We study the following class of starlike functions $$\mathcal{S}^*_{\wp}:=\left\{f\in\mathcal{A}: {zf'(z)}/{f(z)}\prec\ 1+ze^z=:\wp(z) \right\},$$ that are associated with the cardioid domain $\wp(\mathbb{D})$, by deriving certain convolution results, radius problems, majorization result, radius problems in terms of coefficients and differential subordination implications. Consequently, we establish some interesting generalizations of our results for the Ma-Minda class of starlike functions $\mathcal{S}^{*}(\psi)$. We also provide, the set of extremal functions maximizing $\Re\Phi\left(\log{(f(z)/z)}\right)$ or $\left|\Phi\left(\log{(f(z)/z)}\right)\right|$ for functions in $\mathcal{S}^{*}(\psi)$, where $\Phi$ is a non-constant entire function. Further T. H. MacGregor's result for the class $\mathcal{S}^{*}(\alpha)$ and $\mathcal{S}^*_{\wp}$ are obtained as special case to our result

Subclasses of Multivalent Meromorphic Functions with a Pole of Order p at the Origin

In this paper, we carry out a systematic study to discover the properties of a subclass of meromorphic starlike functions defined using the Mittag–Leffler three-parameter function. Differential operators involving special functions have been very useful in extracting information about the various properties of functions belonging to geometrically defined function classes. Here, we choose the Prabhakar function (or a three parameter Mittag–Leffler function) for our study, since it has several applications in science and engineering problems. To provide our study with more versatility, we define our class by employing a certain pseudo-starlike type analytic characterization quasi-subordinate to a more general function. We provide the conditions to obtain sufficient conditions for meromorphic starlikeness involving quasi-subordination. Our other main results include the solution to the Fekete–Szegő problem and inclusion relationships for functions belonging to the defined function classes. Several consequences of our main results are pointed out