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On the order of starlikeness of the class UST. (English) Zbl 0931.30005

Let A denote the class of functions f(z)=z+ k=2 a k (f)z k , regular and normalized in the unit disk D. The author studies the relation between two subclasses of A, the class of starlike functions of order α, α<1,

ST(α)={fA:Re[zf ' (z)/(f(z)]α,zD}

and the class of uniformly starlike functions

UST={fA:Re[(z-ζ)f ' (z)/(f(z)-f(ζ))]0,(z,ζ)D×D}·

F. Rønnig [J. Math. Anal. Appl. 194, No. 1, 319-327 (1995; Zbl 0834.30011)] showed that USTST(1 2) and posed the problem of determining the largest α0 such that USTST(α). The author proved that if α>α 0 =0·1483, then USTST(α). The bound is determined as α 0 =1-h 0 -1/2 , where h 0 =1·3786 is the maximum of the function

h(s,t)=1 41 + s t + (1-s 2 )(1-t 2 )1 + s t + (1-t 2 )(1+t 2 +2st)

in the square 0s,t1 which is attained for s 0 =0·9246, t 0 =0·7803.

30C45Special classes of univalent and multivalent functions