the aim of this paper is to prove some inequalities for p-valent meromorphic functions in thepunctured unit disk δ* and find important corollaries.

It is clear that 0∈ h(Δ). Moreover, (i) if 0 ∈ h(Δ), then h is called spirallike (resp., starlike) with respect to an interior point; (ii) if 0 ∈ h(Δ), then h is called spirallike (resp., starlike) with respect to a boundary point. In this case, there is a boundary point (say, z = 1) such that h(1) := ∠ limz→1h(z)= 0 (see, e.g., [1, 6]); by symbol∠ lim, we denote the angular (nontangential) lim...

A normalized univalent function f is called Ma–Minda starlike or convex if zf ðzÞ=f ðzÞ uðzÞ or 1þ zf ðzÞ=f 0ðzÞ uðzÞ where u is a convex univalent function with uð0Þ 1⁄4 1. The class of Ma–Minda convex functions is shown to be closed under certain operators that are generalizations of previously studied operators. Analogous inclusion results are also obtained for subclasses of starlike and clo...

In the present paper, we introduce and investigate some properties of two subclasses $ Lambda_{n}( lambda , beta ) $ and $ Lambda_{n}^{+}( lambda , beta ) $;  meromorphic and starlike  functions of order $ beta $. In particular, several inclusion relations, coefficient estimates, distortion theorems and covering theorems are proven here for each of these function classes.

The purpose of the present paper is to give some characterizations for a Gaussian hypergeometric function to be in various subclasses of starlike and convex functions. We also consider an integral operator related to the hypergeometric function.

The objective of the present paper is to give some characterizations for a (Gaussian) hypergeometric function to be in various subclasses of starlike and convex functions. We also consider an integral operator related to the hypergeometric function.


 Assume that \(f\) lies in the class of starlike functions order \(\alpha\in[0,1)\), is, which are regular and univalent for \(|z|<1\) such that
 Re\(\left(\frac{zf'(z)}{f(z)}\right)>\alpha\) \(|z|<1.\)
 In this paper we show each following sharp inequality holds:
 \(|f(re^{i\theta})|^{-1}\int_{0}^{r}|f'(ue^{i\theta})|\,du\leq\frac{\Gamma(\frac{1}{2})\Gamma(2-\alph...

Alexander [1] was the first to introduce certain subclasses of univalent functions examining the geometric properties of the image f(D) of D under f . The convex functions are those that map D onto a convex set. A function w = f(z) is said to be starlike if, together with any of its points w, the image f(D) contains the entire segment {tw : 0 ≤ t ≤ 1}. Thus we introduce the denotations S = {f ∈...

Motivated by the recent work on symmetric domains, this article investigates certain features of domain which are caused secant hyperbolic functions. Geometric characteristics analytic functions associated with discussed, include inclusion results, structural formula, sharp radii results such as radius starlikeness and convexity order α. It also finds a for ratios Euler numbers.