Estimations of the second coefficient of a univalent, bounded, symmetric and non-vanishing function by means of Loewner's parametric method

• Autorzy: J. Śladkowska
• Języki publikacji: EN

Abstrakty

EN

Let $𝓑₀^{(R)}(b)$ denote the class of functions F(z) = b + A₁z + A₂z² + ...$ analytic and univalent in the unit disk U which satisfy the conditions: F(U) ⊂ U, 0 ∉ F(U), $Im F^{(n)}(0) = 0$. Using Loewner's parametric method we obtain lower and upper bounds of A₂ in $𝓑₀^{(R)}(b)$ and functions for which these bounds are realized. The class $𝓑₀^{(R)}(b)$, introduced in [6], is a subclass of the class $𝓑_u$ of bounded, non-vanishing univalent functions in the unit disk. This last class and closely related ones have been studied by various authors in [1]-[4]. We mention in particular the paper of D. V. Prokhorov and J. Szynal [5], where a sharp upper bound for the second coefficient in $𝓑_u$ is given.

Słowa kluczowe

EN

univalent function; Loewner differential equation

Kategorie tematyczne

• 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1998
• Tom: 68
• Numer: 2
• Strony: 119-123

Daty

wydano

1998

otrzymano

1996-11-05

poprawiono

1997-02-15

Twórcy

autor

J. Śladkowska

• Institute of Mathematics, Silesian Technical University, Ul. Kaszubska 23, 44-100 Gliwice, Poland

Bibliografia

• [1] P. Duren and G. Schober, Nonvanishing univalent functions, Math. Z. 170 (1980), 195-216.
• [2] C. Horowitz, Coefficients of nonvanishing functions in $H^∞$, Israel J. Math. 30 (1978), 285-291.
• [3] J. Hummel, S. Scheinberg and L. Zalcman, A coefficient problem for bounded nonvanishing functions, J. Anal. Math. 31 (1977), 169-190.
• [4] J. Krzyż, Coefficient problem for bounded nonvanishing functions, Ann. Polon. Math. 70 (1968), 314.
• [5] D. V. Prokhorov and J. Szynal, Coefficient estimates for bounded nonvanishing functions, Bull. Acad. Polon. Sci. Sér. Sci. Math. 29 (1981), 223-230.
• [6] J. Śladkowska, On univalent, bounded, non-vanishing and symmetric functions in the unit disk, Ann. Polon. Math. 64 (1996), 291-299.
• [7] O. Tammi, Extremum Problems for Bounded Univalent Functions, Lecture Notes in Math. 646, Springer, 1978.