In this paper we consider the class of functions f analytic in the open unit disc <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$|z|<1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>z</mi> <mo stretchy="false">|</mo> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> with the normalization <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$f(0)=f'(0)-1=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> that satisfy the condition <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ zf'(z)=G(z)P(z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>z</mi> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>G</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mi>P</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for some odd function G and some function P with positive real part and with certain restrictions on they Taylor’s coefficients. For this class we establish the bounds for the coefficients and for the Fekete-Szegö functional. Moreover, the sharp bound of the second Hankel determinant is found.

Second Hankel determinant and coefficient problems in a set of analytic functions

期刊论文
发表日期：

2026年3月27日

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Edyta Trybucka

摘要

In this paper we consider the class of functions f analytic in the open unit disc $|z|<1$ $| z | < 1$ with the normalization $f(0)=f'(0)-1=0$ $f (0) = f^{'} (0) - 1 = 0$ that satisfy the condition $ zf'(z)=G(z)P(z)$ $z f^{'} (z) = G (z) P (z)$ for some odd function G and some function P with positive real part and with certain restrictions on they Taylor’s coefficients. For this class we establish the bounds for the coefficients and for the Fekete-Szegö functional. Moreover, the sharp bound of the second Hankel determinant is found.