In this paper, we introduce two classes of exponential-type means <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Phi _{p}(a,b;\omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Φ</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>;</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Psi _{p}(a,b;\omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Ψ</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>;</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for distinct positive real numbers a, b with real parameter p and weight <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\omega \in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ω</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. By virtue of these exponential-type means, we establish sharp bounds for the Toader mean <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$T(a,b)=\frac{2}{\pi }\int _{0}^{{\pi }/{2}}\sqrt{a^2{\cos ^2{t}}+b^2{\sin ^2{t}}}dt$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mfrac> <mn>2</mn> <mi>π</mi> </mfrac> <msubsup> <mo>∫</mo> <mrow> <mn>0</mn> </mrow> <mrow> <mi>π</mi> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msubsup> <msqrt> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> <mrow> <msup> <mo>cos</mo> <mn>2</mn> </msup> <mi>t</mi> </mrow> <mo>+</mo> <msup> <mi>b</mi> <mn>2</mn> </msup> <mrow> <msup> <mo>sin</mo> <mn>2</mn> </msup> <mi>t</mi> </mrow> </mrow> </msqrt> <mi>d</mi> <mi>t</mi> </mrow> </math></EquationSource> </InlineEquation>, precisely, for any real number p, we determine the best possible constants <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\omega _i=\omega _i(p)\in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ω</mi> <mi>i</mi> </msub> <mo>=</mo> <msub> <mi>ω</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq6"> <EquationSource Format="TEX">$i=1,2,3,4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>) such that the double inequalities <Equation ID="Equ39"> <EquationSource Format="TEX">$\begin{aligned} \Phi _p\left( a,b;\omega _1\right)<T(a,b)<\Phi _p\left( a,b;\omega _2\right) ,\quad \Psi _p\left( a,b;\omega _3\right)<T(a,b)<\Psi _p\left( a,b;\omega _4\right) \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi mathvariant="normal">Φ</mi> <mi>p</mi> </msub> <mfenced close=")" open="("> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>;</mo> <msub> <mi>ω</mi> <mn>1</mn> </msub> </mfenced> <mo><</mo> <mi>T</mi> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mo><</mo> <msub> <mi mathvariant="normal">Φ</mi> <mi>p</mi> </msub> <mfenced close=")" open="("> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>;</mo> <msub> <mi>ω</mi> <mn>2</mn> </msub> </mfenced> <mo>,</mo> <mspace width="1em" /> <msub> <mi mathvariant="normal">Ψ</mi> <mi>p</mi> </msub> <mfenced close=")" open="("> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>;</mo> <msub> <mi>ω</mi> <mn>3</mn> </msub> </mfenced> <mo><</mo> <mi>T</mi> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mo><</mo> <msub> <mi mathvariant="normal">Ψ</mi> <mi>p</mi> </msub> <mfenced close=")" open="("> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>;</mo> <msub> <mi>ω</mi> <mn>4</mn> </msub> </mfenced> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>hold for all positive <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$a\ne b$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>≠</mo> <mi>b</mi> </mrow> </math></EquationSource> </InlineEquation>. The sharp bounds obtained for the Toader mean generalize several recently established results. As direct applications, we derive several new asymptotic inequalities for the Legendre’s complete elliptic integral of the second kind, which compare favorably to some previous results.

Sharp bounds for the Toader mean by two classes of exponential-type means with applications

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发表日期：

2026年4月7日

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Tiehong Zhao,
Miaokun Wang

摘要

In this paper, we introduce two classes of exponential-type means $\Phi _{p}(a,b;\omega )$ $Φ_{p} (a, b; ω)$ and $\Psi _{p}(a,b;\omega )$ $Ψ_{p} (a, b; ω)$ for distinct positive real numbers a, b with real parameter p and weight $\omega \in (0,1)$ $ω \in (0, 1)$ . By virtue of these exponential-type means, we establish sharp bounds for the Toader mean $T(a,b)=\frac{2}{\pi }\int _{0}^{{\pi }/{2}}\sqrt{a^2{\cos ^2{t}}+b^2{\sin ^2{t}}}dt$ $T (a, b) = \frac{2}{π} \int_{0}^{π / 2} \sqrt{a^{2} {cos}^{2} t + b^{2} {sin}^{2} t} d t$ , precisely, for any real number p, we determine the best possible constants $\omega _i=\omega _i(p)\in (0,1)$ $ω_{i} = ω_{i} (p) \in (0, 1)$ ( $i=1,2,3,4$ $i = 1, 2, 3, 4$ ) such that the double inequalities $\begin{aligned} \Phi _p\left( a,b;\omega _1\right)<T(a,b)<\Phi _p\left( a,b;\omega _2\right) ,\quad \Psi _p\left( a,b;\omega _3\right)<T(a,b)<\Psi _p\left( a,b;\omega _4\right) \end{aligned}$ $\begin{matrix} Φ_{p} (a, b; ω_{1}) < T (a, b) < Φ_{p} (a, b; ω_{2}), Ψ_{p} (a, b; ω_{3}) < T (a, b) < Ψ_{p} (a, b; ω_{4}) \end{matrix}$ hold for all positive $a\ne b$ $a \neq b$ . The sharp bounds obtained for the Toader mean generalize several recently established results. As direct applications, we derive several new asymptotic inequalities for the Legendre’s complete elliptic integral of the second kind, which compare favorably to some previous results.