<p>The main subject of interest of this paper is the problem of estimating <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\vert A_9\vert $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi>A</mi> <mn>9</mn> </msub> <mrow> <mo stretchy="false">|</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, which is the modulus of the ninth coefficient of the inverse of a convex function belonging to the class <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {K}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">K</mi> </math></EquationSource> </InlineEquation>. It was shown almost 50 years ago that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\vert A_n\vert $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi>A</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">|</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$n\ge 10$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>10</mn> </mrow> </math></EquationSource> </InlineEquation>, can exceed 1. On the other hand, it is known that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\vert A_n\vert \le 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi>A</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">|</mo> <mo>≤</mo> <mn>1</mn> </mrow> </mrow> </math></EquationSource> </InlineEquation> for <i>n</i> ranging from 2 to 8. Until now, the problem of finding a sharp bound of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\vert A_9\vert $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi>A</mi> <mn>9</mn> </msub> <mrow> <mo stretchy="false">|</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> has been unsolved. This problem is solved in this paper. Some related problems are also formulated.</p>

Ninth coefficient of the inverse of convex functions

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发表日期：

2026年4月7日

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Paweł Zaprawa

摘要

The main subject of interest of this paper is the problem of estimating $\vert A_9\vert $ $| A_{9} |$ , which is the modulus of the ninth coefficient of the inverse of a convex function belonging to the class $\mathcal {K}$ $K$ . It was shown almost 50 years ago that $\vert A_n\vert $ $| A_{n} |$ , where $n\ge 10$ $n \geq 10$ , can exceed 1. On the other hand, it is known that $\vert A_n\vert \le 1$ $| A_{n} | \leq 1$ for n ranging from 2 to 8. Until now, the problem of finding a sharp bound of $\vert A_9\vert $ $| A_{9} |$ has been unsolved. This problem is solved in this paper. Some related problems are also formulated.