‎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|

For a constant $alphain left(-frac{pi}{2},frac{pi}{2}right)$,  we definea  subclass of the spirallike functions, $SP_{p}(alpha)$, the setof all functions $fin mathcal{A}$[releft{e^{-ialpha}frac{zf'(z)}{f(z)}right}geqleft|frac{zf'(z)}{f(z)}-1right|.]In  the present paper, we shall give the estimate of the norm of the pre-Schwarzian derivative  $mathrm{T}...

Quantitative estimates are obtained for the (finite) valence of functions analytic in the unit disk with Schwarzian derivative that is bounded or of slow growth. A harmonic mapping is shown to be uniformly locally univalent with respect to the hyperbolic metric if and only if it has finite Schwarzian norm, thus generalizing a result of B. Schwarz for analytic functions. A numerical bound is obt...

Over the last decades it has been widely recognized that many completely integrable PDE’s from mathematical physics arise naturally in geometry. Their integrable character in the geometric context – usually associated with the presence of a Lax pair and a spectral deformation – is nothing but the flatness condition of a naturally occurring connection. Examples of interest to geometers generally...

In recent years, a sufficient condition for determining chaotic behaviours of the nonlinear systems has been characterized by the negative Schwarzian derivative (Hacıbekiroǧlu et al, Nonlinear Anal.: Real World Appl. 10, 1270 (2009)). In this work, the Schwarzian derivative has been calculated for investigating the quantum chaotic transition points in the high-temperature superconducting frame ...

Let the functions q1 be analytic and let q2 be analytic univalent in the unit disk. Using the methods of differential subordination and superordination, sufficient conditions involving the Schwarzian derivative of a normalized analytic function f are obtained so that either q1 z ≺ zf ′ z /f z ≺ q2 z or q1 z ≺ 1 zf ′′ z /f ′ z ≺ q2 z . As applications, sufficient conditions are determined relati...

In this paper, we consider a subclass SQ of normalized analytic functions f satisfying ℜf′(z)>1/2. For the in class SQ, determine upper bounds for number coefficient estimates, among which are initial coefficients, second Hankel determinant, and Zalcman functional. Upper estimates higher-order Schwarzian derivatives also obtained.