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On distortion properties of analytic operators. (English) Zbl 0759.30007

The author continues his studies of integral and differential operators, defined on certain families of functions, holomorphic in the open unit disc [see e.g. the author, ibid. 10, 20-38 (1987; Zbl 0614.30029)]. For \(a>0\) let \({\mathcal L}(a)\) be the integral operator, defined by \({\mathcal L}(a)f(z)=a \int_ I t^{a-2} f(zt)dt\), \(I\) the unit interval. The inverse of \({\mathcal L}(a)\) is the differential operator \(\Theta(a)={1\over a}({d \over{d\log z}}+a-1)\). A related one, \(\Lambda(\alpha)=1+\alpha{d\over{d\log z}}\), is earlier discussed by Miller and Altintas. As proved earlir by the author, the operator \({\mathcal L}(a)\) may be interpolated into a family \(\{{\mathcal L}(a)^ \lambda\}\), \(\lambda\geq 0\), satisfying \({\mathcal L}(a)^ \lambda{\mathcal L}(a)^ \mu ={\mathcal L}(a)^{\lambda+\mu}\), where \({\mathcal L}(a)\) may be represented by \[ {\mathcal L}(a)^ \lambda f(z)={a^ \lambda \over \Gamma(\lambda)} \int_ I t^{a-2}\left(\log {1\over t}\right)^{\lambda-1} f(zt)dt. \] Extension to negative \(\lambda\)-values are made. Also for the other operators similar generalizations exist.
Properties of the operators and relations between them are proved. The main part of the paper is devoted to distorsion theorems showing the effect of the operator on \({\mathfrak R}(f(z)/z)\) when \(f(z)\) is transformed. In particular certain functions with an integral representation w.r.t. a complex-valued measure \(\tau\) are studied. Illustrations are given in which \(\tau\) has a direct spectrum, and also applications in form of brief proofs of theorems due to Miller and Altintas, and of an improved form of a result by Chichra.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)

Citations:

Zbl 0614.30029
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References:

[1] ALTINTAS, O., On the coefficients of functions majorized by univalent functions. Hacetteppe Bull. Nat. Sci.-Engrg. 10 (1981), 23-30. · Zbl 0585.30018
[2] CHICHRA, P. N., New subclasses of the class of close-to-convex functions. Proc Amer. Math. Soc. 62 (1977), 37-43. · Zbl 0355.30013 · doi:10.2307/2041942
[3] KOMATU, Y., On a one-parameter additive family of operators defined on analytic functions regular in the unit disk. Bull. Fac. Sci.-Engrg., Chuo Univ. 22 (1979), 1-22. · Zbl 0462.30023
[4] KOMATU, Y., On a family of integral operators related to fractional calculus Kodai Math. J. 10 (1987), 20-38. · Zbl 0614.30029 · doi:10.2996/kmj/1138037357
[5] KOMATU, Y., On distortion properties of operators defined on analytic functions Compl. Anal. Appl. ’87 Sofia 1989, 298-303.
[6] KOMATU, Y., On analytic prolongation of a family of operators. Proc. Coloq Compl. Anal, and Romanian-Finnish Sem. ’89 Bucharest (to appear). · Zbl 0753.30005
[7] KOMATU, Y., On an analytic differential operator. Univalent Functions, Frictional Calculus, and Their Applications. Ellis Horwood Ltd., UK (1989), 103-112. · Zbl 0692.30027
[8] MILLER, S., Differential inequalities and Caratheodory functions. Bull. Amer Math. Soc. 81 (1975), 79-81. · Zbl 0302.30003 · doi:10.1090/S0002-9904-1975-13643-3
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