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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.

For $ a, b > 0 $ with $ a\neq b $, the Gauss lemniscate mean $ \mathcal{LM}(a, b) $ is defined by $ \begin{equation*} \mathcal{LM}(a,b) = \left\{\begin{array}{lll} \frac{\sqrt{a^2-b^2}}{\left[{ {\rm{arcsl}}}\left(\sqrt[4]{1-b^2/a^2}\right)\right]^2}, \ &a>b,\\ \frac{\sqrt{b^2-a^2}}{\left[{ {\rm{arcslh}}}\left(\sqrt[4]{b^2/a^2-1}\right)\right]^2},\ &a

In recent decades, polymer composites are widely used in industry due to their good mechanical properties and their low specific weight. Also, the use of glass fibers and carbon nanotubes can strengthen and improve the mechanical performance of the polymer due to their good mechanical properties. In this study, incorporated glass/epoxy nanocomposite with carbon nanotubes (CNT) samples were fabricated using a hand lay-up process, and the effect of addition functionalized Single-Walled Carbon Nanotubes (F-SWCNT) with COOH and non-functionalized SWCNT was investigated. The tensile strength, elastic modulus, bending strength was obtained experimentally using SANTAM-STM50. X-Ray Diffraction (XRD) and Scanning Electron Microscopy (SEM) were used to investigate the phase and morphology of the fibers. The mechanical properties results showed that the highest elastic modulus and tensile strength are obtained for the sample reinforced with F-SWCNT which increased by 32% and 10%, respectively in comparison with pure epoxy. Also, the obtained results of the bending test indicate that the highest flexural modulus and the highest flexural strength are related to the sample reinforced with functionalized carbon nanotubes which are 16.9 GPa and 381.39 MPa, respectively. Then, the mechanical performance of the reinforcement in the epoxy matrix and the failure mechanism was monitored using SEM images. Finally, reinforced epoxy nanocomposites with functionalized and non-functionalized SWCNT were simulated using Molecular Dynamics (MD) simulation to examine the agreement with the trends of experimental results. The MD obtained results showed that the most appropriate mode of dispersion occurs when functionalized carbon nanotubes are used. Also, it was observed that the elastic modulus of incorporated nanocomposites with F-SWCNT increased by 17% compared to non-functionalized SWCNT which shows the agreement with the trends of experimental results.

Abstract In the article, we provide the sharp bounds for the Sándor–Yang mean in terms of certain families of the two-parameter geometric and arithmetic mean and the one-parameter geometric and harmonic means. As applications, we present new bounds for a certain Yang mean and the inverse tangent function.

Abstract In the article, we prove that λ1=1/2+[(2+log(1+2))/2]1/ν−1/2 $\lambda _{1}=1/2+\sqrt{ [ (\sqrt{2}+ \log (1+\sqrt{2}) )/2 ]^{1/\nu }-1}/2$, μ1=1/2+6ν/(12ν) $\mu _{1}=1/2+\sqrt{6 \nu }/(12\nu )$, λ2=1/2+[(π+2)/4]1/ν−1/2 $\lambda _{2}=1/2+\sqrt{ [(\pi +2)/4 ] ^{1/\nu }-1}/2$ and μ2=1/2+3ν/(6ν) $\mu _{2}=1/2+\sqrt{3\nu }/(6\nu )$ are the best possible parameters on the interval [1/2,1] $[1/2, 1]$ such that the double inequalities Cν[λ1x+(1−λ1)y,λ1y+(1−λ1)x]A1−ν(x,y)0 $x, y>0$ with x≠y $x\neq y$ and ν∈[1/2,∞) $\nu \in [1/2, \infty )$, where A(x,y) $A(x, y)$ is the arithmetic mean, C(x,y) $C(x, y)$ is the contraharmonic mean, and RQA(x,y) $\mathcal{R}_{QA}(x, y)$ and RAQ(x,y) $\mathcal{R}_{AQ}(x, y)$ are two Neuman means.

Abstract In the article, we provide new bounds for two Sándor–Yang means in terms of the arithmetic and contraharmonic means. Our results are the improvements of the previously known results.

Abstract In the article, we provide several sharp upper and lower bounds for two Sándor–Yang means in terms of combinations of arithmetic and contra-harmonic means.

Abstract In the article, we present the best possible parameters λ = λ ( p ) $\lambda=\lambda (p)$ and μ = μ ( p ) $\mu=\mu(p)$ on the interval [ 0 , 1 / 2 ] $[0, 1/2]$ such that the double inequality G p [ λ a + ( 1 − λ ) b , λ b + ( 1 − λ ) a ] A 1 − p ( a , b ) < E ( a , b ) < G p [ μ a + ( 1 − μ ) b , μ b + ( 1 − μ ) a ] A 1 − p ( a , b ) $$\begin{aligned}& G^{p}\bigl[\lambda a+(1-\lambda)b, \lambda b+(1-\lambda)a \bigr]A^{1-p}(a,b) \\& \quad< E(a,b) < G^{p}\bigl[\mu a+(1-\mu)b, \mu b+(1-\mu)a \bigr]A^{1-p}(a,b) \end{aligned}$$ holds for any p ∈ [ 1 , ∞ ) $p\in[1, \infty)$ and all a , b > 0 $a, b>0$ with a ≠ b $a\neq b$ , where A ( a , b ) = ( a + b ) / 2 $A(a, b)=(a+b)/2$ , G ( a , b ) = a b $G(a,b)=\sqrt{ab}$ and E ( a , b ) = [ 2 ∫ 0 π / 2 a cos 2 θ + b sin 2 θ d θ / π ] 2 $E(a,b)=[2\int_{0}^{\pi /2}\sqrt{a\cos^{2}\theta+b\sin^{2}\theta}\,d\theta/\pi]^{2}$ are the arithmetic, geometric and special quasi-arithmetic means of a and b, respectively.

Abstract In the article, we prove that the double inequality x 2 + p 0 x + p 0 < Γ ( x + 1 ) < x 2 + 9 / 5 x + 9 / 5 $$ \frac{x^{2}+p_{0}}{x+p_{0}}< \Gamma(x+1)< \frac{x^{2}+9/5}{x+9/5} $$ holds for all x ∈ ( 0 , 1 ) $x\in(0, 1)$ , we present the best possible constants λ and μ such that λ ( x 2 + 9 / 5 ) x + 9 / 5 ≤ Γ ( x + 1 ) ≤ μ ( x 2 + p 0 ) x + p 0 $$ \frac{\lambda(x^{2}+9/5)}{x+9/5}\leq\Gamma(x+1)\leq\frac{\mu (x^{2}+p_{0})}{x+p_{0}} $$ for all x ∈ ( 0 , 1 ) $x\in(0, 1)$ , and we find the value of x ∗ $x^{\ast}$ in the interval ( 0 , 1 ) $(0, 1)$ such that Γ ( x + 1 ) > ( x 2 + 1 / γ ) / ( x + 1 / γ ) $\Gamma(x+1)>(x^{2}+1/\gamma)/(x+1/\gamma)$ for x ∈ ( 0 , x ∗ ) $x\in(0, x^{\ast})$ and Γ ( x + 1 ) < ( x 2 + 1 / γ ) / ( x + 1 / γ ) $\Gamma(x+1)

Abstract In this paper, we prove that the double inequalities α N Q A ( a , b ) + ( 1 − α ) G ( a , b ) < T D [ A ( a , b ) , G ( a , b ) ] < β N Q A ( a , b ) + ( 1 − β ) G ( a , b ) , λ N A Q ( a , b ) + ( 1 − λ ) G ( a , b ) < T D [ A ( a , b ) , G ( a , b ) ] < μ N A Q ( a , b ) + ( 1 − μ ) G ( a , b ) $$\begin{aligned}& {\alpha }N_{QA}(a,b)+({1-\alpha })G(a,b)< TD \bigl[A(a,b),G(a,b) \bigr]< {\beta }N_{QA}(a,b)+({1-\beta })G(a,b), \\& {\lambda }N_{AQ}(a,b)+({1-\lambda })G(a,b)< TD \bigl[A(a,b),G(a,b) \bigr]< {\mu }N_{AQ}(a,b)+({1-\mu })G(a,b) \end{aligned}$$ hold for all a , b > 0 $a,b>0$ with a ≠ b $a\neq b$ if and only if α ≤ 3 / 8 $\alpha \leq 3/8$ , β ≥ 4 / [ π ( log ( 1 + 2 ) + 2 ) ] = 0.5546 ⋯ $\beta \geq 4/ [\pi ( \log (1+\sqrt{2})+\sqrt{2}) ]=0.5546 \cdots $ , λ ≤ 3 / 10 $\lambda \leq 3/10$ and μ ≥ 8 / [ π ( π + 2 ) ] = 0.4952 ⋯ $\mu \geq 8/ [\pi (\pi +2) ]=0.4952 \cdots $ , where T D ( a , b ) $TD(a,b)$ , G ( a , b ) $G(a,b)$ , A ( a , b ) $A(a,b)$ and N Q A ( a , b ) $N_{QA}(a,b)$ , N A Q ( a , b ) $N_{AQ}(a,b)$ are the Toader, geometric, arithmetic and two Neuman means of a and b, respectively.

Abstract In the article, we provide a monotonicity rule for the function [ P ( x ) + A ( x ) ] / [ P ( x ) + B ( x ) ] $[P(x)+A(x)]/[P(x)+B(x)]$ , where P ( x ) $P(x)$ is a positive differentiable and decreasing function defined on ( − R , R ) $(-R, R)$ ( R > 0 $R>0$ ), and A ( x ) = ∑ n = n 0 ∞ a n x n $A(x)=\sum ^{\infty}_{n=n_{0}}a_{n}x^{n}$ and B ( x ) = ∑ n = n 0 ∞ b n x n $B(x)=\sum^{\infty }_{n=n_{0}}b_{n}x^{n}$ are two real power series converging on ( − R , R ) $(-R, R)$ such that the sequence { a n / b n } n = n 0 ∞ $\{a_{n}/b_{n}\}_{n=n_{0}}^{\infty}$ is increasing (decreasing) with a n 0 / b n 0 ≥ ( ≤ ) 1 $a_{n_{0}}/b_{n_{0}}\geq(\leq)\ 1$ and b n > 0 $b_{n}>0$ for all n ≥ n 0 $n\geq n_{0}$ . As applications, we present new bounds for the complete elliptic integral E ( r ) = ∫ 0 π / 2 1 − r 2 sin 2 t d t $\mathcal{E}(r)=\int_{0}^{\pi /2}\sqrt{1-r^{2}\sin^{2}t}\,dt$ ( 0 < r < 1 $0< r

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.

Abstract In the article, we prove that the double inequalities M α ( a , b ) < S Q A ( a , b ) < M β ( a , b ) $M_{\alpha }(a,b)< S_{QA}(a,b)< M_{\beta}(a,b)$ and M λ ( a , b ) < S A Q ( a , b ) < M μ ( a , b ) $M_{\lambda }(a,b)< S_{AQ}(a,b)< M_{\mu}(a,b)$ hold for all a , b > 0 $a, b>0$ with a ≠ b $a\neq b$ if and only if α ≤ log 2 / [ 1 + log 2 − 2 log ( 1 + 2 ) ] = 1.5517 … $\alpha\leq\log 2/[1+\log2-\sqrt{2}\log(1+\sqrt{2})]=1.5517\ldots$ , β ≥ 5 / 3 $\beta\geq5/3$ , λ ≤ 4 log 2 / [ 4 + 2 log 2 − π ] = 1.2351 … $\lambda\leq4\log2/[4+2\log2-\pi]=1.2351\ldots$ and μ ≥ 4 / 3 $\mu\geq4/3$ , where S Q A ( a , b ) = A ( a , b ) e Q ( a , b ) / M ( a , b ) − 1 $S_{QA}(a,b)=A(a,b)e^{Q(a,b)/M(a,b)-1}$ and S A Q ( a , b ) = Q ( a , b ) e A ( a , b ) / T ( a , b ) − 1 $S_{AQ}(a,b)=Q(a,b)e^{A(a,b)/T(a,b)-1}$ are the Sándor-type means, A ( a , b ) = ( a + b ) / 2 $A(a,b)=(a+b)/2$ , Q ( a , b ) = ( a 2 + b 2 ) / 2 $Q(a,b)=\sqrt{(a^{2}+b^{2})/2}$ , T ( a , b ) = ( a − b ) / [ 2 arctan ( ( a − b ) / ( a + b ) ) ] $T(a,b)=(a-b)/[2\arctan((a-b)/(a+b))]$ , and M ( a , b ) = ( a − b ) / [ 2 sinh − 1 ( ( a − b ) / ( a + b ) ) ] $M(a,b)=(a-b)/[2\sinh ^{-1}((a-b)/(a+b))]$ are, respectively, the arithmetic, quadratic, second Seiffert, and Neuman-Sándor means.

We present the best possible parameters α1,α2,β1,β2∈R and α3,β3∈(1/2,1) such that the double inequalities α1A(a,b)+(1-α1)C(a,b)0 with a≠b and give several sharp inequalities involving the hyperbolic and inverse hyperbolic functions. Here, N(a,b), A(a,b), Q(a,b), and C(a,b) are, respectively, the Neuman, arithmetic, quadratic, and centroidal means of a and b, and NQA(a,b)=N[Q(a,b),A(a,b)].

We present the best possible parameters α1,β1,α2,β2∈R and α3,β3∈(1/2,1) such that the double inequalities Qα1(a,b)A1-α1(a,b)0 with a≠b, where A(a,b), Q(a,b), and AG(a,b) are the arithmetic, quadratic, and Gauss arithmetic-geometric means of a and b, respectively. As applications, we find several new bounds for the complete elliptic integrals of the first and second kind.

We present the best possible parameters λ1,μ1∈R and λ2,μ2∈1/2,1 such that double inequalities λ1C(a,b)+1-λ1A(a,b)0 with a≠b, where A(a,b)=(a+b)/2, C(a,b)=a3+b3/a2+b2 and T(a,b)=2∫0π/2a2cos2θ+b2sin2θdθ/π are the arithmetic, second contraharmonic, and Toader means of a and b, respectively.

We present the sharp bounds for the Neuman means SHA, SAH, SCA and SAC in terms of the arithmetic, harmonic, and contraharmonic means. Our results are the refinements or improvements of the results given by Neuman.

We present several new sharp bounds for Neuman-Sándor mean in terms of arithmetic, centroidal, quadratic, harmonic root square, and contraharmonic means.

We give the greatest values r1, r2 and the least values s1, s2 in (1/2, 1) such that the double inequalities C(r1a+(1-r1)b,r1b+(1-r1)a)0 with a≠b, where A(a,b), M(a,b), C(a,b), and T(a,b) are the arithmetic, Neuman-Sándor, contraharmonic, and second Seiffert means of a and b, respectively.

We present the greatest value p such that the inequality P(a,b)>Lp(a,b) holds for all a,b>0 with a≠b, where P(a,b) and Lp(a,b) denote the Seiffert and pth generalized logarithmic means of a and b, respectively.

We present the largest values α1, α2, and α3 and the smallest values β1, β2, and β3 such that the double inequalities α1M(a,b)+(1-α1)H(a,b)0 with a≠b, where M(a,b), A(a,b), He(a,b), H(a,b) and H-(a,b) denote the Neuman-Sándor, arithmetic, Heronian, harmonic, and harmonic root-square means of a and b, respectively.

We present sharp upper and lower generalized logarithmic mean bounds for the geometric weighted mean of the geometric and harmonic means.

We prove that αH(a,b)+(1−α)L(a,b)>M(1−4α)/3(a,b) for α∈(0,1) and all a,b>0 with a≠b if and only if α∈[1/4,1) and αH(a,b)+(1−α)L(a,b)

In this paper, by proving some monotonicity theorems of certain combinations of the arc lemniscate sine function and elementary functions, we obtain two classes of exponential type inequalities for the arc lemniscate sine function. As applications, sharp bounds for the lemniscatic mean in terms of the arithmetic, harmonic and geometric means are given, which extend some previously known results. [ABSTRACT FROM AUTHOR]