Let An(p)(p,n ∈ N = {1,2,3,...}) denote the class of functions of the form f(z) = zp +ap+nzp+n +...which are analytic in the unit disk E = {z:|z| < 1}.A function f(z) in A(p) is said to be in the class Sp(a,b) if ...Let An(p)(p,n ∈ N = {1,2,3,...}) denote the class of functions of the form f(z) = zp +ap+nzp+n +...which are analytic in the unit disk E = {z:|z| < 1}.A function f(z) in A(p) is said to be in the class Sp(a,b) if it satisfies zf(z) pf(z) 1 + az 1 + bz for some a and b(-1 ≤ b < a ≤ 1).In this paper,using the method of differential subordinations,we give new criteria for f(z) to be in the classes Sp(a,b)(-1 ≤ b < a ≤ 1),that is,if f(z) ∈ A(p) satisfies f(z) = 0 in 0 < |z| < 1,f(z) = 0 when μ = 1,and zf(z) f(z) μ 1 + zff((zz)) + λzff(z(z))h(z),where h(z) = p1+μ(1 + λ) 11 ++ abzz 1+μ + pμ(1 + az)1-μ(1+bz)1+μ,λ≥-1,μ∈ {-1,0,1},then f(z) ∈ Sp(a,b).Our results improve or extend some results due to Owa,Nunokawa,Padmanabhan,Silverman,Obradovic,Yang and others[3-10].展开更多
基金Supported by the Natural Science Foundation of Jiangsu Education Department（06KJD110003）
文摘Let An(p)(p,n ∈ N = {1,2,3,...}) denote the class of functions of the form f(z) = zp +ap+nzp+n +...which are analytic in the unit disk E = {z:|z| < 1}.A function f(z) in A(p) is said to be in the class Sp(a,b) if it satisfies zf(z) pf(z) 1 + az 1 + bz for some a and b(-1 ≤ b < a ≤ 1).In this paper,using the method of differential subordinations,we give new criteria for f(z) to be in the classes Sp(a,b)(-1 ≤ b < a ≤ 1),that is,if f(z) ∈ A(p) satisfies f(z) = 0 in 0 < |z| < 1,f(z) = 0 when μ = 1,and zf(z) f(z) μ 1 + zff((zz)) + λzff(z(z))h(z),where h(z) = p1+μ(1 + λ) 11 ++ abzz 1+μ + pμ(1 + az)1-μ(1+bz)1+μ,λ≥-1,μ∈ {-1,0,1},then f(z) ∈ Sp(a,b).Our results improve or extend some results due to Owa,Nunokawa,Padmanabhan,Silverman,Obradovic,Yang and others[3-10].
