Scholarlink

<p>In this article, we discuss the coefficients problems for Bloch functions. A general theorem on the sharp estimate of the weighted sum of the absolute values of squares of coefficients of Bloch functions is proved. Using this theorem, for fixed <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(0&lt;r\le 1/\sqrt{3},\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>&lt;</mo> <mi>r</mi> <mo>≤</mo> <mn>1</mn> <mo stretchy="false">/</mo> <msqrt> <mn>3</mn> </msqrt> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> we improve a result of I.R.&#xa0;Kayumov and K.-J. Wirths (Monat. Math. <b>190</b>, 123–135 (2019)), namely we improve the upper bound for the infimum of the set of numbers <i>a</i>(<i>r</i>) such that the value <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(S_rf-a(r)|f^{\prime }(0)|^2,\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mi>r</mi> </msub> <mi>f</mi> <mo>-</mo> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mrow> <mo stretchy="false">|</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(S_rf\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mi>r</mi> </msub> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation> is the area functional, attains its maximum in the Bloch class at some monomial. The obtained estimate is asymptotically sharp as <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(r\rightarrow 0.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo stretchy="false">→</mo> <mn>0</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

On the weighted sum of squares of the coefficients of Bloch functions

  • 期刊论文

  • 发表日期：

    2026年1月9日

查看全文
  • Ramis Khasianov

摘要

In this article, we discuss the coefficients problems for Bloch functions. A general theorem on the sharp estimate of the weighted sum of the absolute values of squares of coefficients of Bloch functions is proved. Using this theorem, for fixed \(0<r\le 1/\sqrt{3},\) 0 < r ≤ 1 / 3 , we improve a result of I.R. Kayumov and K.-J. Wirths (Monat. Math. 190, 123–135 (2019)), namely we improve the upper bound for the infimum of the set of numbers a(r) such that the value \(S_rf-a(r)|f^{\prime }(0)|^2,\) S r f - a ( r ) | f ′ ( 0 ) | 2 , where \(S_rf\) S r f is the area functional, attains its maximum in the Bloch class at some monomial. The obtained estimate is asymptotically sharp as \(r\rightarrow 0.\) r → 0 .