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

Abstract

Let $B_p(a,;beta,;lambda;;j)$ be the class consisting of functions $f(z)=z^p+{textstylesum_{k=p+1}^infty};a_kz^k,;pin N$ which satisfy $Re{;afrac{f^{(j)}(z)}{z^{p-j}}+betafrac{f^{(j+1)}(z)}{z^{p-3-1}}+(frac{beta-alpha}2);frac{f^{(j+2)}(z)}{z^{p-j-2}}};>lambda$, (z ∈ U = {z : |z| 0 or α = β = 1 . The extreme points of Bp(α, β, λ; j) are determined and various sharp inequalities related to Bp(α, β, λ; 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 λ are found so that functions ofBp(α, β, λ; j) are p-valent starlike in U . For instance, Re{zf ′′(z)} > (5 − 12 ln 2)/(44 − 48 ln 2) ≈ −0.309 is sufficient condition for any normalized analytic function f to be starlike in U . The results improve and include a number of known results as their special cases.