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On bounded univalent functions realizing a local extremum of two coefficients. (Russian) Zbl 0697.30017

Let \(S_ R(M)\), \(M>1\), denote the family of functions \(f(z)=z+A_ 2z^ 2+..\). holomorphic and univalent in the disc \(\Delta =\{z:\) \(| z| <1\}\), satisfying the conditions: \(| f(z)| <M\) for \(z\in \Delta\), \(A_ n=\bar A_ n\) for \(n=2,3,... \). In the paper it is proved that if there exists a function \(w=f_ 0(z)\) for which in the family \(S_ R(M)\) the maxima of the coefficients \(A_ m\) and \(A_ n\), \(m\neq n\), m,n\(\geq 2\), are attained simultaneously, then it satisfies in the disc \(\Delta\) the equation \[ (w/[(\epsilon -(w/M)^{d_ 0})({\bar \epsilon}- (w/M)^{d_ 0}]^{1/d_ 0})=(z/[(\epsilon -z^{d_ 0})({\bar \epsilon}-z^{d_ 0})]^{1/d_ 0}), \] where \(| \epsilon | =1\), \(d_ 0\) some common divisor of (m-1) and (n-1), \(^{d_ 0}\sqrt{1}=1\). The special cases: a) n, \(n+1\), b) \(n=p+1\), \(2\leq m\leq n- 1\), where p is an arbitrary prime number, were considered by the reviewer and W. Majchrzak [Serdica 10, 337-343 (1984; Zbl 0576.30030)]. The respective investigations in the general class S(M) were carried out by S. Śladkowska [Demonstr. Math. 11, 351-378 (1978; Zbl 0393.30011)].
Reviewer: Z.J.Jakubowski

MSC:

30C55 General theory of univalent and multivalent functions of one complex variable
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