Let $f$ be a locally univalent function on the unit disk $U$. We consider the normalized extensions of $f$ to the Euclidean unit ball $B^nsubseteqmathbb{C}^n$ given by $$Phi_{n,gamma}(f)(z)=left(f(z_1),(f'(z_1))^gammahat{z}right),$$  where $gammain[0,1/2]$, $z=(z_1,hat{z})in B^n$ and $$Psi_{n,beta}(f)(z)=left(f(z_1),(frac{f(z_1)}{z_1})^betahat{z}right),$$ in which $betain[0,1]$, $f(z_1)neq 0$ a...

‎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|

the results of mndo, am1 and pm3 semi-empirical calculations and hf/6-31g*, mp/6-31g*, mp2/6-311+g**, and b3lyp/6-311g** and qcisd/6-31g* ab initio methods for angle strain and conjugation effects in twist-boat, boat, and half-chair geometries of (z,z,z)-cycloocta-1,3,5-triene indicate that all methods, except pm3, predict wrong ordering of the conformations.

background & aim: ziziphora l. genus belongs to the lamiaceae family, which has important medicinal and aromatic plant species. in current study, essential oil chemical compositions of two ziziphora l. species (z. tenuior l. and z. clinopodioides lam.) were investigated. experimental: the flowering branches of z. tenuior and z. clinopodioides were collected in spring and summer 2011. the ae...

‎let $p(z)=z^s h(z)$ where $h(z)$ is a polynomial‎ ‎of degree at most $n-s$ having all its zeros in $|z|geq k$ or in $|z|leq k$‎. ‎in this paper we obtain some new results about the dependence of $|p(rz)|$ on $|p(rz)| $ for $r^2leq rrleq k^2$‎, ‎$k^2 leq rrleq r^2$ and for $rleq r leq k$‎. ‎our results refine and generalize certain well-known polynomial inequalities‎.