The well-known Schwarz–Pick lemma states that any analytic mapping φ of the unit disk U into itself satisfies the inequality |φ′(z)| ≤ 1− |φ(z)| 2 1− |z|2 , z ∈ U. This estimate remains the same if we restrict ourselves to univalent mappings. The lower estimate is |φ′(z)| ≥ 0 generally or |φ′(z)| > 0 for univalent functions. To make the lower estimate nontrivial we consider univalent functions ...

in this paper‎, ‎we introduce a new class$t_{k}^{s,a}[a,b,alpha‎ ,‎beta ]$ of analytic functions by using a‎ ‎newly defined convolution operator‎. ‎this class contains many known classes of‎ ‎analytic and univalent functions as special cases‎. ‎we derived some‎ ‎interesting results including inclusion relationships‎, ‎a radius problem and‎ ‎sharp coefficient bound for this class‎.

In the past few years, many scholars gave much attention to use of q-calculus in geometric functions theory, and they defined new subclasses analytic harmonic functions. While using symmetric function very little work has been published so far. this research, with help fundamental concepts q-Salagean differential operator for functions, we define a class connected Janowski SH0˜m,q,A,B. First, i...

The class SH consists of univalent, harmonic, and sense-preserving functions f in the unit disk, ∆, such that f = h + g where h(z) = z + ∑∞ 2 akz , g(z) = ∑∞ 1 bkz . Using a technique from Clunie and Sheil-Small, we construct a family of 1-slit mappings in SH by varying ω(z) = g ′(z)/f ′(z). As ω(z) changes, the tip of the slit slides along the negative real axis from the point 0 to −1. In doin...