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 α1, β1, α2, β2 ∈ ℝ and α3, β3 ∈ [1/2, 1] such that the double inequalities$$\begin{array}{} \begin{split} \displaystyle \alpha_{1}C(a, b)+(1-\alpha_{1})A(a, b) & \lt T_{3}(a, b) \lt \beta_{1}C(a, b)+(1-\beta_{1})A(a, b), \\ \alpha_{2}C(a, b)+(1-\alpha_{2})Q(a, b) & \lt T_{3}(a, b) \lt \beta_{2}C(a, b)+(1-\beta_{2})Q(a, b), \\ C(\alpha_{3}; a, b) & \lt T_{3}(a, b) \lt C(\beta_{3}; a, b) \end{split} \end{array}$$hold for a, b > 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, $\begin{array}{} \displaystyle Q(a, b)=\sqrt{\left(a^{2}+b^{2}\right)/2} \end{array}$ is the quadratic mean, C(a, b) = (a2 + b2)/(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 $\begin{array}{} T_{3}(a,b)=\Big(\frac{2}{\pi}\int\limits_{0}^{\pi/2}\sqrt{a^{3}\cos^{2}\theta+b^{3}\sin^{2}\theta}\text{d}\theta\Big)^{2/3} \end{array}$ is the Toader mean of order 3.

Optimal bounds of classical and non-classical means in terms of Q means

“…decreases, so by Lemma 3 h(z) = z−arsinhz z 3 decreases, assuming values between lim z→0 + h(z) = 1 6 and h(1) = 1 − arsinh1, which completes the proof. Theorem 10 (Bounds for the sine mean).…”

“…The authors wish to thank the anonymous referee for their careful reading of the manuscript. We are grateful for their suggestions to add references [3][4][5][6][7]9,10,14,[16][17][18][19], that present the main research content of the article, and the research background and research progress in this field.…”

Optimal bounds for the sine and hyperbolic tangent means IV

“…If −1/3 ≤ u < 0, we have 1+3u u(u−1) ≥ 0, so g u is convex thus positive. For −1 < u < −1/3, the equation tanh 2 x + 1+3u u(u−1) = 0 has exactly one solution ξ u , so g u is concave and negative on (0, ξ u ). Then it becomes convex and tends to infinity, thus assumes zero at exactly one point x u .…”

“…Proof Using Seiffert's functions we have to proof that h(z) = tanh z − 3z 3+z 2 > 0 for 0 < z < 1. Note that cosh z = 1 + z 2 2! + z 4 4!…”

Performance Optimization of the Helical Heat Exchanger With Turbulator

“…The process of transferring the heat (Ghalandari et al, 2019;Zhao et al, 2021a, Zhao et al, 2021b between two fluids at different temperatures separated by a solid wall is common in many engineering applications (Che et al, 2021;Mahariq and Erciyas, 2017;Mahariq et al, 2020;Panahi and Zamzamian, 2017;Prabhanjan et al, 2002;Shafee et al, 2020;Zhao et al, 2021c). Heat exchangers are devices that allow heat to be transferred from one fluid to another without mixing the two fluids (Assad et al, 2021;Che et al, 2021;Chu et al, 2020;She and Fan, 2018;Zhou et al, 2020). Significant issues such as savings in materials, space, energy, and the global economy have led to the development of more efficient heat exchanger equipment and reductions in costs (Ghalandari et al, 2020;Irandoost Shahrestani et al, 2020;Shadloo et al, 2016;Zhao et al, 2020a).…”