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ON THE CLASS Φp*(α,β,γ,ε,η;A) of p-VALENT FUNCTIONS WITHREFERENCE TO THE BERNARDI INTEGRAL OPERATOR
Jian Ming
Acta mathematica scientia,Series B. 1994, 14 (4):
371-376.
Bernardi has proved that if f(z) is starlike univalent in the unit disk △, then so is the function g(z) given by
g(z)=(c+1)z-1∫0ztc-1f(t)dt,c=1,2….In this paper, we extend Bernardi's theorem to certain class Φp*(α,β,γ,ε,η;A) of p-valent starlike functions for operators. We prove that if f(A)∈Φp*(α,β,γ,ε,η;A), then g(A), defined by
g(A)=(p+c)∫01tc-1f(tA)dt,c=1,2,… also belongs to Φp*(α,β,γ,ε,η;A) and give a sufficient and necessary vondition with reference to f(A) ∈Φp*(α,β,γ,ε,η;A).
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