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Radii of convexity and close-to-convexity of certain integral representations

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Strict upper bounds are determined for ¦s(z)¦, ¦Re s(z)¦, and ¦Im s(z) ¦ in the class of functions s(z)=a nzn+an+1zn+1+... (n1) regular in ¦z¦<1 and satisfying the condition ¦u1) −u2) ¦≤K¦ θ1-θ 2¦, where U(θ)=Re s (ei θ), K>0, andθ 1 andθ 2 are arbitrary real numbers. These bounds are used in the determination of radii of convexity and close-to-convexity of certain integral representations.

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Translated from Matematicheskie Zametki, Vol. 7, No. 5, pp. 581–592, May, 1970.

The author wishes to thank L. A. Aksent'ev for his guidance in this work.

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Avkhadiev, F.G. Radii of convexity and close-to-convexity of certain integral representations. Mathematical Notes of the Academy of Sciences of the USSR 7, 350–357 (1970).

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  • Real Number
  • Integral Representation
  • Arbitrary Real Number