Let A denote the class of funetions f(z) defined by f(z)=z-sum from n=2 to ∞a_nz^n (a_n>0) (1) which are analytic in the unit disk U= {z:|z|<1}. A function f(z)∈A is said to be in the class S (x,β,γ), if and...Let A denote the class of funetions f(z) defined by f(z)=z-sum from n=2 to ∞a_nz^n (a_n>0) (1) which are analytic in the unit disk U= {z:|z|<1}. A function f(z)∈A is said to be in the class S (x,β,γ), if and only if |(f′(z)-1)/(af′(z)+1-(1+α)β)|<γ(2) for 0<x<1, 0<β<1, and 0<<1.展开更多
Suppose that g(z)=z+a_2z^2+…is analytic for |z|<1, The condition Re{z g(z)/g(z)}>0 for |z|<1 (1) is necessary and sufficient for g(z) to be univalent and starlike for |z|<1. The condition Re{z g″(z)/g′(...Suppose that g(z)=z+a_2z^2+…is analytic for |z|<1, The condition Re{z g(z)/g(z)}>0 for |z|<1 (1) is necessary and sufficient for g(z) to be univalent and starlike for |z|<1. The condition Re{z g″(z)/g′(z)+1}>0 for |z|<1 (2) is necessary and sufficient for g(z) to be univalent and convex for |z|<1 In this paper we determine the radius Univalence (and starlikeness) of f(z) assodated with each of the cases (ⅰ) g(z) satisfies Re {g(z)/z}>0 for |z|<1 (ⅱ) g(z) satisfies eq .(1), (ⅲ) g(z) satisfies eq. (2). Definition. Let N_αdenote the class of functions f(z) which are regular in|z|<1 and are normalized by f(0)=1 and map |z|<1 into region D_α=展开更多
文摘Let A denote the class of funetions f(z) defined by f(z)=z-sum from n=2 to ∞a_nz^n (a_n>0) (1) which are analytic in the unit disk U= {z:|z|<1}. A function f(z)∈A is said to be in the class S (x,β,γ), if and only if |(f′(z)-1)/(af′(z)+1-(1+α)β)|<γ(2) for 0<x<1, 0<β<1, and 0<<1.
文摘Suppose that g(z)=z+a_2z^2+…is analytic for |z|<1, The condition Re{z g(z)/g(z)}>0 for |z|<1 (1) is necessary and sufficient for g(z) to be univalent and starlike for |z|<1. The condition Re{z g″(z)/g′(z)+1}>0 for |z|<1 (2) is necessary and sufficient for g(z) to be univalent and convex for |z|<1 In this paper we determine the radius Univalence (and starlikeness) of f(z) assodated with each of the cases (ⅰ) g(z) satisfies Re {g(z)/z}>0 for |z|<1 (ⅱ) g(z) satisfies eq .(1), (ⅲ) g(z) satisfies eq. (2). Definition. Let N_αdenote the class of functions f(z) which are regular in|z|<1 and are normalized by f(0)=1 and map |z|<1 into region D_α=
