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Inclusion theorems for convolution product of second order polylogarithms and functions with the derivative in a halfplane. (English) Zbl 0915.30012

Let 𝒜 denote the family of normalized regular functions defined in the unit disc Δ={z:|z|<1}. For β<1 and real η, let η (β) denote the family of functions f𝒜 such that Re[e iη (f ' (z)-β)]>0 for aΔ. Given a generalized second order polylogarithm function

G(a,b;z)= n=1 (a+1)(b+1) (n+a)(n+b)z n ,a,b-1,-2,-3,,

we place conditions on the parameters a, b and β to guarantee that the Hadamard product of the power series H f (a,b;z)G(a,b;z)*f(z) will be univalent, starlike or convex. We give conditions on a and b to describe the geometric nature of the function G(a,b;z). We note that for f𝒜, the function H f (a,b;z) satisfies the differential equation

z 2 H f '' (z)+(a+b+1)zH f ' (z)+abH f (z)=(a+1)(b+1)f(z),

and H f has the integral representation

H f (a,b;z):=(a+1)(b+1) b-a 0 1 t a-1 (1-t b-a )f(tz)dt,ifbaandH f (a,a;z):=(1+a) 2 0 1 (log1/t)t -1 f(tz)dt,forRea>-1·

If a>-1, and b>a with b, we see that H f (a,b;z) reduces to the well-known Bernardi transform. If a=-α and b=2-α, H f (a,b;z) is the operator considered by R. Ali and V. Singh [Complex Variables, Theory Appl. 26, No. 4, 299-309 (1995; Zbl 0851.30005)] with an additional assumption that 0α<1. Thus, the operator H f (a,b;z) is the natural choice to study its behaviour. By making f in the class of convex functions, we also find a sufficient condition for H f (a,b;z) to belong to the class 0 (β). Several open problems have been raised at the end.

MSC:
30C45Special classes of univalent and multivalent functions
30C80Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable)