Some properties for a class of analytic functions defined by a higher-order differential inequality

Authors: OQLAH ALREFAI

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|<1\}), $ for some $\lambda\;(\lambda0$ or $\alpha=\beta=1$. The extreme points of $\mathcal{B}_p(\alpha,\beta, \lambda;j)$ are determined and various sharp inequalities related to $\mathcal{B}_p(\alpha,\beta, \lambda;j)$ are obtained. These include univalence criteria, coefficient bounds, growth and distortion estimates and bounds for certain linear operators. Furthermore, inclusion properties are investigated and estimates on $\lambda$ are found so that functions of $\mathcal{B}_p(\alpha,\beta, \lambda;j)$ are p-valent starlike in $\mathbb{U}$. For instance, $\mathrm{Re}\{zf''(z)\}> (5-12\ln 2)/(44-48\ln 2)\approx -0.309$ 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

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