Toader Mean Research Articles

Sharp inequalities for Toader mean in terms of other bivariate means

In the paper, the author discovers the best constants $\alpha_1$, $\alpha_2$, $\alpha_3$, $\beta_1$, $\beta_2$ and $\beta_3$ for the double inequalities \[ \alpha_1 A\left(\frac{a-b}{a+b}\right)^{2n+2} < T(a,b)-\frac{1}{4}C-\frac{3}{4}A-A\sum_{k=1}^{n-1}\frac{(\frac{1}{2},k)^2}{4((k+1)!)^2}\left(\frac{a-b}{a+b}\right)^{2k+2}<\beta_1 A\left(\frac{a-b}{a+b}\right)^{2n+2} \] \[ \alpha_2 A\left(\frac{a-b}{a+b}\right)^{2n+2} < T(a,b)-\frac{3}{4}\overline{C}-\frac{1}{4}A-A\sum_{k=1}^{n-1}\frac{(\frac{1}{2},k)^2}{4((k+1)!)^2}\left(\frac{a-b}{a+b}\right)^{2k+2}<\beta_2 A\left(\frac{a-b}{a+b}\right)^{2n+2} \] and \[ \alpha_3 A\left(\frac{a-b}{a+b}\right)^{2n+2} < \frac{4}{5}T(a,b)+\frac{1}{5}H-A-A\sum_{k=1}^{n-1}\frac{(\frac{1}{2},k)^2}{5((k+1)!)^2}\left(\frac{a-b}{a+b}\right)^{2k+2}<\beta_3 A\left(\frac{a-b}{a+b}\right)^{2n+2} \] to be valid for all $a,b>0$ with $a\ne b$ and $n=1,2,\cdots$, where \[ C\equiv C(a,b)=\frac{a^2+b^2}{a+b},\,\overline{C}\equiv\overline{C}(a,b)=\frac{2(a^2+ab+b^2)}{3(a+b)},\, A\equiv A(a,b)=\frac{a+b}{2}, \] \[ H\equiv H(a,b) =\frac{2ab}{a+b},\quad T(a,b)=\frac2{\pi}\int_0^{\pi/2}\sqrt{a^2\cos^2\theta+b^2\sin^2\theta}\,{\rm d}\theta \] are respectively the contraharmonic, centroidal, arithmetic, harmonic and Toader means of two positive numbers $a$ and $b$, $ (a,n)=a(a+1)(a+2)(a+3)\cdots (a+n-1)$ denotes the shifted factorial function. As an application of the above inequalities, the author also find a new bounds for the complete elliptic integral of the second kind.

Sharp inequalities for the Toader mean of order -1 in terms of other bivariate means

Sharp inequalities for the Toader mean of order -1 in terms of other bivariate means

Sharp generalized Seiffert mean bounds for the Toader mean of order 4

In the article, we present the best possible parameters $$\alpha ,\beta $$ such that the double inequality $$\begin{aligned} S_{\alpha }(a,b)<T_{4}(a,b)<S_{\beta }(a,b) \end{aligned}$$ holds for $$a, b>0$$ with $$a\ne b$$ , and provide new bounds for the complete elliptic integral of the second kind, where $$S_{p}(a,b)$$ and $$T_{4}(a,b)$$ are the generalized Seiffert mean and Toader mean of order 4, respectively.

Sharp bounds for the Toader mean in terms of arithmetic and geometric means

For $$a,b>0$$ with $$a\ne b$$ , the Toader mean of a and b is defined by $$\begin{aligned} T\left( a,b\right) =\frac{2}{\pi }\int _{0}^{\pi /2}\sqrt{a^{2}\cos ^{2}t+b^{2}\sin ^{2}t}dt. \end{aligned}$$ In this paper, we prove that the double inequality $$\begin{aligned} \left( \frac{3}{2}\left( \frac{a+b}{2}\right) ^{q}-\frac{1}{2}\left( \sqrt{ab }\right) ^{q}\right) ^{1/q}<T\left( a,b\right) <\left( \frac{3}{2}\left( \frac{a+b}{2}\right) ^{p}-\frac{1}{2}\left( \sqrt{ab}\right) ^{p}\right) ^{1/p} \end{aligned}$$ holds if and only if $$0<p\le 3/2$$ and $$q\ge \ln \left( 3/2\right) /\ln \left( 4/\pi \right) $$ . This gives new sharp lower and upper bounds for the Toader mean, and improves several known results.

Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means

Abstract In the article, we present the best possible parameters α 1, β 1, α 2, β 2 ∈ ℝ and α 3, β 3 ∈ [1/2, 1] such that the double inequalities α 1 C ( a , b ) + ( 1 − α 1 ) A ( a , b ) &lt; T 3 ( a , b ) &lt; β 1 C ( a , b ) + ( 1 − β 1 ) A ( a , b ) , α 2 C ( a , b ) + ( 1 − α 2 ) Q ( a , b ) &lt; T 3 ( a , b ) &lt; β 2 C ( a , b ) + ( 1 − β 2 ) Q ( a , b ) , C ( α 3 ; a , b ) &lt; T 3 ( a , b ) &lt; C ( β 3 ; a , b ) $$\begin{array}{} \begin{split} \displaystyle \alpha_{1}C(a, b)+(1-\alpha_{1})A(a, b) &amp; \lt T_{3}(a, b) \lt \beta_{1}C(a, b)+(1-\beta_{1})A(a, b), \\ \alpha_{2}C(a, b)+(1-\alpha_{2})Q(a, b) &amp; \lt T_{3}(a, b) \lt \beta_{2}C(a, b)+(1-\beta_{2})Q(a, b), \\ C(\alpha_{3}; a, b) &amp; \lt T_{3}(a, b) \lt C(\beta_{3}; a, b) \end{split} \end{array}$$ hold for a, b &gt; 0 with a ≠ b, and provide new bounds for the complete elliptic integral of the second kind, where A(a, b) = (a + b)/2 is the arithmetic mean, Q ( a , b ) = a 2 + b 2 / 2 $\begin{array}{} \displaystyle Q(a, b)=\sqrt{\left(a^{2}+b^{2}\right)/2} \end{array}$ is the quadratic mean, C(a, b) = (a 2 + b 2)/(a + b) is the contra-harmonic mean, C(p; a, b) = C[pa + (1 – p)b, pb + (1 – p)a] is the one-parameter contra-harmonic mean and T 3 ( a , b ) = ( 2 π ∫ 0 π / 2 a 3 cos 2 ⁡ θ + b 3 sin 2 ⁡ θ d θ ) 2 / 3 $\begin{array}{} T_{3}(a,b)=\Big(\frac{2}{\pi}\int\limits_{0}^{\pi/2}\sqrt{a^{3}\cos^{2}\theta+b^{3}\sin^{2}\theta}\text{d}\theta\Big)^{2/3} \end{array}$ is the Toader mean of order 3.

Optimal bounds for Toader mean in terms of general means

In this paper, we present the best possible parameters alpha (r), beta (r) such that the double inequality [α(r)Mr(a,b)+(1−α(r))Nr(a,b)]1/r<TD[M(a,b),N(a,b)]<[β(r)Mr(a,b)+(1−β(r))Nr(a,b)]1/r,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} {}[\\alpha (r)M^{r}(a,b)+(1-\\alpha (r))N^{r}(a,b)] ^{1/r} < &\\operatorname{TD}\\bigl[M(a,b),N(a,b)\\bigr] \\\\ < &\\bigl[\\beta (r)M^{r}(a,b)+\\bigl(1-\\beta (r)\\bigr)N^{r}(a,b) \\bigr]^{1/r}, \\end{aligned}$$ \\end{document} holds for all rleq 1 and a,b>0 with aneq b, where TD(a,b):=∫0π/2a2cos2θ+b2sin2θdθ\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\operatorname{TD}(a,b):= \\int ^{\\pi /2}_{0}\\sqrt{a^{2}\\cos ^{2}\\theta +b^{2}\\sin ^{2} \\theta }\\,d\\theta $$\\end{document} is the Toader mean, and M, N are means. As applications, we attain the optimal bounds for the Toader mean in terms of arithmetic, contraharmonic, centroidal and quadratic means, and then we provide some new bounds for the complete elliptic integral of the second kind.

Bounds for Toader Mean in Terms of Arithmetic and Second Seiffert Means

In the article, we prove that the double inequalities \begin{align*} &\alpha_{1}T(a,b)+(1-\alpha_{1})A(a,b) 0\) with \(a\neq b\) if and only if \(\alpha_{1}\leq 3/4\), \(\beta_{1}\geq1\), \(\alpha_{2}\leq 3/4\) and \(\beta_{2}\geq 1\), where \(A(a,b)\), \(TD(a,b)\) and \(T(a,b)\) are the arithmetic, Toader and second Seiffert means of \(a\) and \(b\), respectively.

On Approximating the Toader Mean by Other Bivariate Means

In the article, we provide several sharp bounds for the Toader mean by use of certain combinations of the arithmetic, quadratic, contraharmonic, and Gaussian arithmetic-geometric means.

Optimal bounds for arithmetic-geometric and Toader means in terms of generalized logarithmic mean

In this paper, we find the greatest values alpha_{1},alpha_{2} and the smallest values beta_{1},beta_{2} such that the double inequalities L_{alpha_{1}}(a,b)<operatorname{AG}(a,b)<L_{beta_{1}}(a,b) and L_{alpha_{2}}(a,b)< T(a,b)< L_{beta_{2}}(a,b) hold for all a, b>0 with aneq b, where operatorname{AG}(a,b), T(a,b) and L_{p}(a,b) are the arithmetic-geometric, Toader and generalized logarithmic means of two positive numbers a and b, respectively.

Optimal inequalities for bounding Toader mean by arithmetic and quadratic means

In this paper, we present the best possible parameters alpha(r) and beta(r) such that the double inequality \t\t\t[α(r)Ar(a,b)+(1−α(r))Qr(a,b)]1/r<TD[A(a,b),Q(a,b)]<[β(r)Ar(a,b)+(1−β(r))Qr(a,b)]1/r\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} \\bigl[\\alpha(r)A^{r}(a,b)+ \\bigl(1-\\alpha(r) \\bigr)Q^{r}(a,b) \\bigr]^{1/r} < & TD \\bigl[A(a,b), Q(a,b) \\bigr] \\\\ < & \\bigl[\\beta(r)A^{r}(a,b)+ \\bigl(1-\\beta(r) \\bigr)Q^{r}(a,b) \\bigr]^{1/r} \\end{aligned}$$ \\end{document} holds for all rleq 1 and a, b>0 with aneq b, and we provide new bounds for the complete elliptic integral mathcal{E}(r)=int_{0}^{pi/2}(1-r^{2}sin^{2}theta)^{1/2},dtheta(rin (0, sqrt{2}/2)) of the second kind, where TD(a,b)=frac{2}{pi}int_{0}^{pi/2}sqrt{a^{2}cos^{2}theta+b^{2}sin^{2}theta},dtheta, A(a,b)=(a+b)/2 and Q(a,b)=sqrt{(a^{2}+b^{2})/2} are the Toader, arithmetic, and quadratic means of a and b, respectively.

Bounds for the arithmetic mean in terms of the Toader mean and other bivariate means

Bounds for the arithmetic mean in terms of the Toader mean and other bivariate means

A double inequality for the combination of Toader mean and the arithmetic mean in terms of the contraharmonic mean

We find the greatest value ? and the least value ? such that the double inequality C(?a +(1-?)b, ?b + (1-?)a) &lt; ?A(a,b) + (1-?)T(a, b)&lt; C(?a + (1-?)b, ?b + (1-?)a) holds for all ? ? (0,1) and a, b &gt; 0 with a ? b, where C(a,b), A(a,b), and T(a,b) denote respectively the contraharmonic, arithmetic, and Toader means of two positive numbers a and b.

Sharp bounds for Toader mean in terms of arithmetic, quadratic, and Neuman means

In this paper, we present the best possible parameters $\alpha, \beta \in\mathbb{R}$ and $\lambda, \mu\in(1/2, 1)$ such that the double inequalities $\alpha N_{AQ}(a,b)+(1-\alpha)A(a,b)< T^{\ast}(a,b)<\beta N_{AQ}(a,b)+(1-\beta)A(a,b)$ , $Q[\lambda a+(1-\lambda)b, \lambda b+(1-\lambda)a]< T^{\ast}(a,b)< Q[\mu a+(1-\mu)b, \mu b+(1-\mu)a] $ hold for all $a, b>0$ with $a\neq b$ , where $T^{\ast}(a,b)$ , $A(a,b)$ , $Q(a,b)$ and $N_{QA}(a,b)$ are the Toader, arithmetic, quadratic, and Neuman means of a and b, respectively.

Sharp Bounds for Toader Mean in terms of Arithmetic and Second Contraharmonic Means

We present the best possible parametersλ1,μ1∈Randλ2,μ2∈1/2,1such that double inequalitiesλ1C(a,b)+1-λ1A(a,b)&lt;T(a,b)&lt;μ1C(a,b)+1-μ1A(a,b),Cλ2a+1-λ2b,λ2b+1-λ2a&lt;T(a,b)&lt;Cμ2a+1-μ2b,μ2b+1-μ2ahold for alla,b&gt;0witha≠b, whereA(a,b)=(a+b)/2,C(a,b)=a3+b3/a2+b2andT(a,b)=2∫0π/2a2cos2θ+b2sin2θdθ/πare the arithmetic, second contraharmonic, and Toader means ofaandb, respectively.

A double inequality for bounding Toader mean by the centroidal mean

In the paper, the authors find the best numbers $\alpha$ and $\beta$ such that $$ \overline{C}\bigl(\alpha a+(1-\alpha)b,\alpha b+(1-\alpha)a\bigr)<T(a,b) <\overline{C}\bigl(\beta a+(1-\beta)b,\beta b+(1-\beta)a\bigr) $$ for all $a,b>0$ with $a\ne b$, where $\overline{C}(a,b)={2\bigl(a^2+ab+b^2\bigr)}{3(a+b)}$ and $T(a,b)=\frac{2}{\pi}\int_{0}^{{\pi}/{2}}\sqrt{a^2{\cos^2{\theta}}+b^2{\sin^2{\theta}}}\,d\theta$ denote respectively the centroidal mean and Toader mean of two positive numbers $a$ and $b$.

The best bounds for Toader mean in terms of the centroidal and arithmetic means

In the paper, the authors discover the best constants ?1, ?2, ?1, and ?2 for the double inequalities ?1C(a,b) + (1-?1)A(a,b) &lt; T(a,b) &lt; ?1C(a,b) + (1-?1)A(a,b) and ?2/A(a,b) + 1-?2/C(a,b) &lt; 1/T(a,b) &lt; ?2/A(a,b) + 1-?2-C(a,b) to be valid for all a, b &gt; 0 with a ? b, where C(a,b) = 2(a2+ab+b2)/3(a+b), A(a,b) = a+b/2, and T(a,b) = 2/???2,0 ?a2 cos2 ? + b2 sin2 ? d ? are respectively the centroidal, arithmetic, and Toader means of two positive numbers a and b. As an application of the above inequalities, the authors also find some new bounds for the complete elliptic integral of the second kind.

Bounds for Combinations of Toader Mean and Arithmetic Mean in Terms of Centroidal Mean

The authors find the greatest value λ and the least value μ, such that the double inequality holds for all α ∈ (0,1) and a, b > 0 with a ≠ b, where , A(a, b) = (a + b)/2, and denote, respectively, the centroidal, arithmetic, and Toader means of the two positive numbers a and b.

Some Inequalities for Bounding Toader Mean

By finding linear relations among differences between two special means, the authors establish some inequalities for bounding Toader mean in terms of the arithmetic, harmonic, centroidal, and contraharmonic means.

Optimal bounds for Toader mean in terms of arithmetic and contraharmonic means

We find the greatest value α1 and α2, and the least values β1 and β2, such that the double inequalities α1C(a,b)+(1 − α1)A(a,b) 0 with ab. As applications, we get new bounds for the complete elliptic integral of the second kind. Here, C(a,b )=( a 2 +b 2 )/(a+b), A(a,b )=( a+b)/2, and

Sharp bounds for Toader mean in terms of contraharmonic mean with applications

We find the greatest value λ and the least value μ in (1/2,1) such that the dou- ble inequality C(λa+(1 − λ)b,λb+(1 − λ)a) 0 with ab, and give new bounds for the perimeter of an ellipse. Here, T(a,b )= 2 π/2 0 a 2cos2 θ +b2sin 2 θdθ ,a ndC(a,b )=( a 2 +b 2 )/(a+b) denote the Toader, and contraharmonic means of two positive numbers a and b, respectively.