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In this paper, we provide a systematic way to study on some general Wilker-Huygens type inequalities for the trigonometric and hyperbolic functions, lemniscate and hyperbolic lemniscate functions, and their corresponding inverse functions. Our results are some extensions and refinements of the recently published results in [A. Mhanna, On a general Huygens-Wilker inequality, Appl. Math. E.-Notes, 20 (2020), 79-81; MR4076436], and improve many previous results involving Wilker-Huygens type inequalities.

This paper deals with the arc lemniscate functions from the point view of bivariate means which have been introduced in [1]. In this study, several optimal bounds for these bivariate means in terms of arithmetic, geometric and quadratic means are established. As a consequence, new bounds for the arc lemniscate functions are also derived, which improve some previously known results.

In the article, we present the sharp upper and lower bounds for the arithmetic mean in terms of new Seiffert-like means, which give some refinements of the results obtained in [ 1 ]. As applications, two new inequalities for the sine and hyperbolic sine functions will be established.

In the article, we present the best possible bounds for the weighted Holder mean of the zero-balanced generalized complete elliptic integrals of the first and second kinds, which are the generalizations of previous results for complete p-elliptic integrals.

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 ) < T 3 ( a , b ) < β 1 C ( a , b ) + ( 1 − β 1 ) A ( a , b ) , α 2 C ( a , b ) + ( 1 − α 2 ) Q ( a , b ) < T 3 ( a , b ) < β 2 C ( a , b ) + ( 1 − β 2 ) Q ( a , b ) , C ( α 3 ; a , b ) < T 3 ( a , b ) < C ( β 3 ; a , b ) $$\begin{array}{} \begin{split} \displaystyle \alpha_{1}C(a, b)+(1-\alpha_{1})A(a, b) & \lt T_{3}(a, b) \lt \beta_{1}C(a, b)+(1-\beta_{1})A(a, b), \\ \alpha_{2}C(a, b)+(1-\alpha_{2})Q(a, b) & \lt T_{3}(a, b) \lt \beta_{2}C(a, b)+(1-\beta_{2})Q(a, b), \\ C(\alpha_{3}; a, b) & \lt T_{3}(a, b) \lt C(\beta_{3}; a, b) \end{split} \end{array}$$ 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, 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.

In the article, we present an elegant double inequality for the ratio of the zero-balanced hypergeometric functions, which improve and refine some previously known results and also give a positive answer the question by proposed by Ismail.

In the article, we establish a sharp double inequality involving the ratio of generalized complete elliptic integrals of the first kind, which is the improvement and generalization of some previously known results.

In this paper, we present sharp power mean bounds for the so-called lemniscate type means, which were introduced by Neuman (Math Pannon 18(1):77–94, 2007). The obtained results measure what the distance is between the lemniscate-type means and power means. As applications, several new bounds for the arc lemniscate functions are established.

In the article, we present the best possible parameters $$\alpha ,\beta $$ such that the double inequality $$\begin{aligned} S_{\alpha }(a,b)0$$ 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.

Singular ruled surface is an interesting research object and is the breakthrough point of exploring new problems. However, because of singularity, it’s difficult to study the properties of singular ruled surfaces. In this paper, we combine singularity theory and Clifford algebra to study singular ruled surfaces. We take advantage of the dual number of Clifford algebra to make the singular ruled surfaces transform into the dual singular curves on the dual unit sphere. By using the research method on the singular curves, we give the definition of the dual evolute of the dual front in the dual unit sphere, we further provide the k-th dual evolute of the dual front. Moreover, we consider the ruled surface corresponding to the dual evolute and k-th dual evolute and provide the developable conditions of these ruled surfaces.

In this paper, we present the monotonicity properties of the ratio between generalized elliptic integral of the first kind $${\mathcal {K}}_a(r)$$ and its approximation $$\log [1+2/(ar')]$$ , and also the convexity (concavity) of their difference for $$a\in (0,1/2]$$ . As an application, we give new bounds for generalized Grotzsch ring function $$\mu _a(r)$$ and a upper bound for $${\mathcal {K}}_a(r)$$ .

In the article, we present several new monotonicity properties and bounds involving the generalized Grötzsch ring functions $\mu _{a,b}$μa,b in the theory of Ramanujan’s generalized modular equation for $0< a, b0

In this paper, we establish a necessary and sufficient condition for the convexity and concavity of the modified Bessel functions of the first kind with respect to Holder means.

In this paper, we present the best possible parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha(r)$\end{document}α(r) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\beta(r)$\end{document}β(r) such that the double inequality \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \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}[α(r)Ar(a,b)+(1−α(r))Qr(a,b)]1/r0$\end{document}a,b>0 with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a\neq b$\end{document}a≠b, and we provide new bounds for the complete elliptic integral \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{E}(r)=\int_{0}^{\pi/2}(1-r^{2}\sin^{2}\theta)^{1/2}\,d\theta$\end{document}E(r)=∫0π/2(1−r2sin2θ)1/2dθ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(r\in (0, \sqrt{2}/2))$\end{document}(r∈(0,2/2)) of the second kind, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$TD(a,b)=\frac{2}{\pi}\int_{0}^{\pi/2}\sqrt{a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta}\,d\theta$\end{document}TD(a,b)=2π∫0π/2a2cos2θ+b2sin2θdθ, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A(a,b)=(a+b)/2$\end{document}A(a,b)=(a+b)/2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$Q(a,b)=\sqrt{(a^{2}+b^{2})/2}$\end{document}Q(a,b)=(a2+b2)/2 are the Toader, arithmetic, and quadratic means of a and b, respectively.

The current article explores the effect of viscous dissipation on the hybrid nanofluid flow over a stretching sheet. Further, different nanoparticles can be used to enhance the thermal conductivity of the fluid. The resultant suspension is examined with the effect of velocity slip. The velocity slip effect is incorporated in the boundary condition of the momentum equation. Heat transfer analysis is done in presence of convective conditions and thermal radiation. Comparative study of different hybrid nanofluids and nanofluid is considered. Here we have chosen manganese Zinc ferrite MnZnFe2O4 and Nickle Zinc ferrite NiZnFe2O4 as nanoparticles in base fluids Kerosene oil and engine oil. Mathematical modeling of the present flow system is illustrated and solved by the shooting built-in method. Graphs for velocity, temperature, skin friction and Nusselt number are presented for different parameters. Here we noticed that velocity profile of MnZnF e 2 O 4 − NiZnF e 2 O 4 − C 1 0 H 2 2 − C 8 H 18 is maximum as compared to other hybrid nanofluids. Velocity and temperature have opposite effects against slip parameter. The temperature of the fluid rises with Biot number and thermal radiation.

In the article, we present several quadratic transformation inequalities for Gaussian hypergeometric function and find the analogs of duplication inequalities for the generalized Grötzsch ring function.

In this paper, we find the greatest values \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha_{1},\alpha_{2}$\end{document}α1,α2 and the smallest values \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\beta_{1},\beta_{2}$\end{document}β1,β2 such that the double inequalities \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L_{\alpha_{1}}(a,b)0$\end{document}a,b>0 with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a\neq b$\end{document}a≠b, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\operatorname{AG}(a,b)$\end{document}AG(a,b), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T(a,b)$\end{document}T(a,b) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L_{p}(a,b)$\end{document}Lp(a,b) are the arithmetic-geometric, Toader and generalized logarithmic means of two positive numbers a and b, respectively.

In this article, we present several best possible lower bounds for the Neuman-Sandor mean in terms of the geometric combinations of harmonic and quadratic means, geometric and quadratic means, harmonic and contraharmonic means, and geometric and contraharmonic means. Mathematics subject classification (2010): 26E60.

In this paper, we present the best possible parameters $p, q\in\mathbb {R}$ such that the double inequality $M_{p}(a,b)< T[A(a,b), Q(a,b)]< M_{q}(a,b)$ holds for all $a, b>0$ with $a\neq b$ , and we get sharp bounds for the complete elliptic integral $\mathcal{E}(t)=\int _{0}^{\pi/2}(1-t^{2}\sin^{2}\theta)^{1/2}\,d\theta$ of the second kind on the interval $(0, \sqrt{2}/2)$ , where $T(a,b)=\frac{2}{\pi }\int _{0}^{\pi/2}\sqrt{a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta}\,d\theta$ , $A(a,b)=(a+b)/2$ , $Q(a,b)=\sqrt{(a^{2}+b^{2})/2}$ , $M_{r}(a,b)=[(a^{r}+b^{r})/2]^{1/r}$ ( $r\neq0$ ), and $M_{0}(a,b)=\sqrt {ab}$ are the Toader, arithmetic, quadratic, and rth power means of a and b, respectively.