On fractional derivatives of univalent functions

Univalent Functions with Univalent Derivatives. Ii

Univalent Functions Defined by Ruscheweyh Derivatives

We study some radii problems concerning the integral operator z F(z)y+l uY-I f(u) du zy o for certain classes, namely K and M (a), of univalent functions defined by Ruscheweyh n n derivatives. Infact, we obtain the converse of Ruscheweyh’s result and improve a result of Goel and Sohi for complex by a different technique. The results are sharp.

Some Distortion Inequalities Associated with the Fractional Derivatives of Analytic and Univalent Functions

For the classes S and K of (normalized) univalent and convex analytic functions, respectively, a number of authors conjectured interesting extensions of certain known distortion inequalities in terms of a fractional derivative operator. While examining and investigating the validity of these conjectures, many subsequent works considered various generalizations of the distortion inequalities rel...

The norm of pre-Schwarzian derivatives on bi-univalent functions of order $alpha$

‎In the present investigation‎, ‎we give the best estimates for the norm of the pre-Schwarzian derivative $ T_{f}(z)=dfrac{f^{''}(z)}{f^{'}(z)} $ for bi-starlike functions and a new subclass of bi-univalent functions of order $ alpha $‎, ‎where‎ ‎$Vert T_{f} Vert= sup_{|z|

On Univalent Harmonic Functions

Two classes of univalent harmonic functions on unit disc satisfying the conditions ∑∞ n=2(n−α)(|an|+|bn|) ≤ (1−α)(1−|b1|) and ∑∞ n=2 n(n−α)(|an|+|bn|) ≤ (1−α)(1−|b1|) are given. That the ranges of the functions belonging to these two classes are starlike and convex, respectively. Sharp coefficient relations and distortion theorems are given for these functions. Furthermore results concerning th...