- Turkish Journal of Mathematics
- Vol: 43 Issue: 5
- Some properties for a class of analytic functions defined by a higher-order differential inequality
Some properties for a class of analytic functions defined by a higher-order differential inequality
Authors : Oqlah Alrefai
Pages : 2473-2493
View : 8 | Download : 5
Publication Date : 9999-12-31
Article Type : Makaleler
Abstract :Let $\mathcal{B}_p(\alpha,\beta, \lambda;j)$ be the class consisting of functions $f(z)= z^p+\sum_{k=p+1}^{\infty}a_k z^{k},\; p\in \mathbb{N}$ which satisfy $ \mathrm{Re}\left\{\alpha\frac{f^{(j)}(z)}{z^{p-j}}+\beta\frac{f^{(j+1)}(z)}{z^{p-j-1}}+\left(\frac{\beta-\alpha}{2}\right)\frac{f^{(j+2)}(z)}{z^{p-j-2}}\right\}>\lambda,\;\;(z\in \mathbb{U}=\{z:\;|z| (5-12\ln 2)/(44-48\ln 2)\approx -0.309$ is sufficient condition for any normalized analytic function $f$ to be starlike in $\mathbb{U}$. The results improve and include a number of known results as their special cases.Keywords : Starlike functions, p-valent functions, Jack's lemma, univalent functions, extreme points, convex functions, distortion and growth theorem, coefficient bounds, differential inequality