For any holomorphic function f in the Fock space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {F}_\alpha ^2$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="script">F</mi> <mi>α</mi> <mn>2</mn> </msubsup> </math></EquationSource> </InlineEquation>, we give a new representation of the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {F}_\alpha ^2$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="script">F</mi> <mi>α</mi> <mn>2</mn> </msubsup> </math></EquationSource> </InlineEquation> norm in terms of the higher order derivative of f. Furthermore, we prove a sharp inequality <Equation ID="Equ22"> <EquationSource Format="TEX">$\begin{aligned} \int _{\mathbb {C}}\frac{|f^{(n)}(z)|^2e^{-\alpha |z|^2}}{\alpha ^nn!L_n(-\alpha |z|^2)}dA(z) \le \int _{\mathbb {C}}|f(z)|^2e^{-\alpha |z|^2}dA(z), \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mo>∫</mo> <mi mathvariant="double-struck">C</mi> </msub> <mfrac> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <msup> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msup> <mrow> <mi>α</mi> <mo stretchy="false">|</mo> <mi>z</mi> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> </mrow> </msup> </mrow> <mrow> <msup> <mi>α</mi> <mi>n</mi> </msup> <mi>n</mi> <mo>!</mo> <msub> <mi>L</mi> <mi>n</mi> </msub> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mi>α</mi> <mo stretchy="false">|</mo> <mi>z</mi> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </mfrac> <mi>d</mi> <mi>A</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <msub> <mo>∫</mo> <mi mathvariant="double-struck">C</mi> </msub> <msup> <mrow> <mo stretchy="false">|</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msup> <mrow> <mi>α</mi> <mo stretchy="false">|</mo> <mi>z</mi> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> </mrow> </msup> <mi>d</mi> <mi>A</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where n is a positive integer and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> is the Laguerre polynomial. This solves a conjecture posed by Kalaj (Comput Methods Funct Theory 24:283–302, 2024).

A proof of Kalaj’s conjecture

期刊论文
发表日期：

2026年3月5日

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Jineng Dai

摘要

For any holomorphic function f in the Fock space $\mathcal {F}_\alpha ^2$ $F_{α}^{2}$ , we give a new representation of the $\mathcal {F}_\alpha ^2$ $F_{α}^{2}$ norm in terms of the higher order derivative of f. Furthermore, we prove a sharp inequality $\begin{aligned} \int _{\mathbb {C}}\frac{|f^{(n)}(z)|^2e^{-\alpha |z|^2}}{\alpha ^nn!L_n(-\alpha |z|^2)}dA(z) \le \int _{\mathbb {C}}|f(z)|^2e^{-\alpha |z|^2}dA(z), \end{aligned}$ $\begin{matrix} \int_{C} \frac{| f^{(n)} {(z) |}^{2} e^{- {α | z |}^{2}}}{α^{n} n! L_{n} {(- α | z |}^{2})} d A (z) \leq \int_{C} {| f (z) |}^{2} e^{- {α | z |}^{2}} d A (z), \end{matrix}$ where n is a positive integer and $L_n$ $L_{n}$ is the Laguerre polynomial. This solves a conjecture posed by Kalaj (Comput Methods Funct Theory 24:283–302, 2024).