Abstract
Let B-p (alpha, beta, lambda; j) be the class consisting of functions f(z) = z(p) + Sigma(infinity)(k=p+1) a(k)z(k), p is an element of N which satisfy
Re {alpha f((j))(z)/z(p-j) + beta f((j+1))(z)/z(p-j-1 )+ (beta - alpha/2) f((j+2))(z)/z(p-j-2)} > lambda, (z is an element of U = {z : vertical bar z vertical bar < 1}),
for some lambda (lambda < p!{alpha + (p-j)beta+(p-j)(p-j-1)(beta-alpha)/2}/(p-j)!) and j = 0,1, ..., p, where p+1- j+2 alpha/(beta-alpha) > 0 or alpha = beta = 1. The extreme points of B-p(alpha, beta, lambda; j) are determined and various sharp inequalities related to 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 B-p(alpha, beta, lambda; j) are p-valent starlike in U. For instance, Re{z f '' (z)} > (5 - 12 ln 2)/(44 - 48 ln 2) approximate to -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.