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In this paper, we establish an interesting chain of sharp inequalities involving Toader-Qi mean, exponential mean, logarithmic mean, arithmetic mean and geometric mean. This greatly improves some existing results.

This paper focuses on two-point boundary value problem for first order causal difference equations. We will start with two new comparison theorems. Then, by utilizing these theorems and fixed point theorems, we obtain the existence of solutions for the corresponding linear problem. By applying monotone iterative technique, sufficient conditions for the existence of extremal solutions are also established. The results of this paper extend some existing results in the literature. Finally, two examples to show the usefulness of our results are exhibited.

Suppose $w, v>0$, $w\neq v$ and $A_{u}\left (w, v\right) $ is the $u$-order power mean (PM) of $w$ and $v$. In this paper, we completely describe the convexity of $u\mapsto A_{u}\left (w, v\right) $ on $\mathbb{R}$ and $% s\mapsto A_{u\left (s\right) }\left (w, v\right) $ with $u\left (s\right) = \left (\ln 2\right) /\ln \left (1/s\right) $ on $\left (0, \infty \right) $. These yield some new inequalities for PMs, and give an answer to an open problem.

To assess the trends in characteristics, treatments, and outcomes of acute myocardial infarction (AMI) patients in tertiary Chinese medicine (CM) hospitals in China between 2006 and 2013. This retrospective study was based on two nationwide epidemiological surveys of AMI in tertiary CM hospitals during 2 years (2006 and 2013). Patients admitted to the hospital for AMI were enrolled. Hospital records were used as the data source. Case data were derived regarding baseline characteristics, treatments, and outcomes of patients to assess changes from 2006 to 2013. Logistic regression was used to analyze the relationship between prognosis, general influencing factors of disease, and various treatment measures. Totally 26 tertiary CM hospitals in 2006 and 29 tertiary CM hospitals in 2013 (18 were repetitive) were surveyed. A total of 2,311 patients with AMI were enrolled (1,094 cases in 2006 and 1,217 cases in 2013). From 2006 to 2013, the mean age did not significantly change, but the proportion of patients younger than 65 years increased. The prevalence of risk factors such as hypertension, diabetes, and hyperlipidemia also increased. Significant increases were observed in primary percutaneous coronary intervention [20.48% (2006) vs. 24.90% (2013)] and revascularization [36.11% (2006) vs. 52.42% (2013)]. In-hospital mortality decreased from 11.15% in 2006 to 10.60% in 2013. A mortality logistic regression analysis identified reperfusion therapy [odds ratio (OR), 0.222; 95% confidence interval (CI), 0.106–0.464], Chinese patent medicines (OR, 0.394; 95% CI, 0.213–0.727), and CM decoctions (OR, 0.196; 95% CI, 0.109–0.353) as protective factors. Reperfusion and revascularization capabilities of tertiary CM hospitals have improved significantly, but in-hospital mortality has not significantly decreased. Efforts are needed to improve medical awareness of AMI and expand the use of CM to reduce in-hospital mortality in China.

In this paper, we establish a new asymptotic expansion of a ratio of two gamma functions, that is, as x → ∞ x\rightarrow \infty , [ Γ ( x + u ) Γ ( x + v ) ] 1 / ( u − v ) ∼ ( x + σ ) exp ⁡ [ ∑ k = 1 m B 2 n + 1 ( ρ ) w n ( 2 n + 1 ) ( x + σ ) − 2 k + R m ( x ; u , v ) ] , \begin{equation*} \left [ \frac {\Gamma \left ( x+u\right ) }{\Gamma \left ( x+v\right ) }\right ] ^{1/\left ( u-v\right ) }\thicksim \left ( x+\sigma \right ) \exp \left [ \sum _{k=1}^{m}\frac {B_{2n+1}\left ( \rho \right ) }{wn\left ( 2n+1\right ) }\left ( x\!+\!\sigma \right ) ^{-2k}\!+\!R_{m}\left ( x;u,v\right ) \right ] , \end{equation*} where u , v ∈ R u,v\in \mathbb {R} with w = u − v ≠ 0 w=u-v\neq 0 and ρ = ( 1 − w ) / 2 \rho =\left ( 1-w\right ) /2 , σ = ( u + v − 1 ) / 2 \sigma =\left ( u+v-1\right ) /2 , B 2 n + 1 ( ρ ) B_{2n+1}\left ( \rho \right ) are the Bernoulli polynomials. We also prove that the function x ↦ ( − 1 ) m R m ( x ; u , v ) x\mapsto \left ( -1\right ) ^{m}R_{m}\left ( x;u,v\right ) for m ∈ N m\in \mathbb {N} is completely monotonic on ( − σ , ∞ ) \left ( -\sigma ,\infty \right ) if | u − v | > 1 \left \vert u-v\right \vert >1 , which yields an explicit bound for | R m ( x ; u , v ) | \left \vert R_{m}\left ( x;u,v\right ) \right \vert and some new inequalities.

The Smith's approximation formula for the gamma function is given by Γ ( x + 1 2 ) = 2 π ( x e ) x ( 2 x tanh ⁡ 1 2 x ) x / 2 ( 1 + O ( 1 x 5 ) ) , as x → ∞ . In this paper, by means of a little known power series, we develop this formula to an asymptotic expansions: Γ ( x + 1 / 2 ) 2 π ( x / e ) x ∼ ( 2 x tanh ⁡ 1 2 x ) x / 2 exp ⁡ ( ∑ n = 3 ∞ [ ( 2 n ) ! − ( 2 n − 1 ) 2 2 n − 1 ] ( 1 − 2 1 − 2 n ) 2 n ( 2 n − 1 ) ( 2 n ) ! B 2 n x 2 n − 1 ) , as x → ∞ , and give an estimate of remainder in the above asymptotic series, where B 2 n is the Bernoulli number. Moreover, as relevant results, Ramanujan type asymptotic expansions for the gamma function Γ ( x + 1 / 2 ) are obtained; an asymptotic expansion for Wallis ratio and an estimate of the remainder are established.

In this paper, we prove that for $$a\ge 31/98$$, the function $$\begin{aligned} f_{a}\left( x\right) =\ln \Gamma \left( x+\frac{1}{2}\right) -x\ln x+x-\frac{ 1}{2}\ln \left( 2\pi \right) +\dfrac{x}{24}\dfrac{x^{2}+a-7/120}{ x^{4}+ax^{2}+\left( 98a-31\right) /1680} \end{aligned}$$is strictly increasing (decreasing) and concave (convex) on $$\left( 0,\infty \right) $$ if and only if $$a\ge 5281/6068$$ ($$a=31/98$$). Moreover, we show that the necessary and sufficient condition for function $$\begin{aligned} F_{a}\left( x\right) =-\left( x^{4}+ax^{2}+\frac{98a-31}{1680}\right) f_{a}\left( x\right) \end{aligned}$$for $$a\in \mathbb {R}$$ to be completely monotonic on $$\left( 0,\infty \right) $$ is also $$a\ge 5281/6068$$. These yield some new sharp bounds for the gamma function.

Linguistic preference relations (LPRs) can indicate the decision makers (DMs)’ qualitative pairwise judgments regarding a set of alternatives in uncertain multicriteria decision-making problems. This paper examines several goal programming models for managing the additive consistency and consensus of LPRs and then develops an additive-consistency- and consensus-based method for group decision making (GDM) with LPRs. First, this paper offers a consistency index to quantify the consistency level for LPRs and define acceptable consistent LPRs. For an LPR that is unacceptably additive consistent, several additive-consistency-based programming models are developed to address the inconsistency and to establish an acceptably consistent LPR. Then, an additive-consistency-based procedure to generate the priority weight vector from the LPR is offered. An additive-consistency-based algorithm for decision making with an LPR is presented. Subsequently, considering the consensus in GDM, a consensus index is proposed for gauging the agreement degree among individual LPRs. Regarding individual LPRs that do not exhibit acceptably additive consistency or acceptable consensus, several goal programming models to derive new LPRs with acceptable consistency and consensus are provided. Afterward, the DMs’ weights are determined objectively, and individual LPRs are integrated into a collective LPR. An additive-consistency- and consensus-based GDM method with a group of LPRs is developed. Finally, two practical numerical examples are offered, and a comparative analysis is presented.

In the paper, by means of the monotonicity rule for the ratio of two Laplace transform we determine the best $$a,b>0$$ such that the double inequality $$\begin{aligned} \exp \left( -\frac{x}{24x^{2}+b}\right)x_{0}\ge 0$$ . Furthermore, we give some completely monotonic functions involving gamma function, and establish an asymptotic expansion of $$\Gamma \left( x+1/2\right) $$ . Finally, an open problem is proposed.

Let f and g be both continuous functions on ( 0 , ∞ ) with g ( t ) > 0 for t ∈ ( 0 , ∞ ) and let F ( x ) = L ( f ) , G ( x ) = L ( g ) be respectively the Laplace transforms of f and g converging for x > 0 . We prove that if there is a t ⁎ ∈ ( 0 , ∞ ) such that f / g is strictly increasing on ( 0 , t ⁎ ) and strictly decreasing on ( t ⁎ , ∞ ) , then the ratio F / G is decreasing on ( 0 , ∞ ) if and only if H F , G ( 0 + ) = lim x → 0 + ⁡ ( F ′ ( x ) G ′ ( x ) G ( x ) − F ( x ) ) ≥ 0 , with lim x → 0 + ⁡ F ( x ) G ( x ) = lim t → ∞ ⁡ f ( t ) g ( t ) and lim x → ∞ ⁡ F ( x ) G ( x ) = lim t → 0 + ⁡ f ( t ) g ( t ) provided the indicated limits exist. While H F , G ( 0 + ) 0 , there is at least one x ⁎ > 0 such that F / G is increasing on ( 0 , x ⁎ ) and decreasing on ( x ⁎ , ∞ ) . As applications, a unified treatment for certain bounds of psi function is presented, and some properties of the modified Bessel functions of the second are established. These show that the monotonicity rules in this paper may contribute to study for certain special functions because many special functions can be expressed as corresponding Laplace transforms.

In the paper, by virtue of two new tools, the authors present the monotonicity and convexity of certain combinations of the complete elliptic integrals of the first kind, and obtain new sharp bounds and inequalities for them.

Let $$\psi _{n}=\left( -1\right) ^{n-1}\psi ^{\left( n\right) }$$ for $$n\ge 0$$ , where $$\psi ^{\left( n\right) }$$ stands for the psi and polygamma functions. For $$p,q\in \mathbb {R}$$ and $$\rho =\min \left( p,q\right) $$ , let $$\begin{aligned} D\left[ x+p,x+q;\psi _{n-1}\right] \equiv -\phi _{n}\left( x\right) \end{aligned}$$ be the divided difference of the functions $$\psi _{n-1}$$ for $$x>-\rho $$ . In this paper, we prove that for $$\left| p-q\right| \gtrless 1$$ , the sequences $$\{\xi _{n}\left( x\right) \}_{n\in \mathbb {N}}$$ and $$\{\eta _{n}\left( x\right) \}_{n\in \mathbb {N}}$$ defined by $$\begin{aligned} \xi _{n}\left( x\right) =\frac{n\phi _{n}\left( x\right) }{\phi _{n+1}\left( x\right) }-x\text { and }\eta _{n}\left( x\right) =\left[ \frac{\phi _{n}\left( x\right) }{\left( n-1\right) !}\right] ^{-1/n}-x \end{aligned}$$ are strictly decreasing (increasing) and converge to $$\min \left( p,q\right) $$ . Furthermore, if $$\left| p-q\right| \gtrless 1$$ , then the functions $$x\mapsto \xi _{n}\left( x\right) $$ and $$x\mapsto \eta _{n}\left( x\right) $$ are strictly increasing (decreasing) from $$\left( -\rho ,\infty \right) $$ onto $$\left( \rho ,\left( p+q-1\right) /2\right) $$ ( $$\left( \left( p+q-1\right) /2,\rho \right) $$ ). These not only generalize and strengthen some known results, but also yield several new sharp bounds involving polygamma functions.

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}

Inspired by Ramanujan’s unusual approximations for the perimeter of an ellipse with semiaxis 1 and $$r\in \left( 0,1\right) $$ , a family of approximations of Ramanujan type is constructed. We find the sharp lower and upper approximations have very high accuracy, and better than some known ones.

For u , v , r , s ∈ R and 0 q ≠ 1 , let Γ q , ψ q be the q-gamma, -psi functions and let W q ; u , v be defined on ( − min ⁡ { u , v } , ∞ ) by W q ; u , v ( x ) = ( Γ q ( x + u ) Γ q ( x + v ) ) 1 / ( u − v ) if u ≠ v and W q ; u , u ( x ) = e ψ q ( x + u ) . In this paper, by a new way we present the necessary and sufficient conditions for the ratio ( W q ; u , v / W q ; r , s ) to be logarithmically completely monotonic on ( − ρ , ∞ ) , where ρ = min ⁡ { u , v , r , s } . This extends and generalizes some existing results.

Let $${\mathcal {K}}\left( r\right) $$ be the complete elliptic integrals of the first kind and $$\text{ arth }r$$ denote the inverse hyperbolic tangent function. We prove that the inequality $$\begin{aligned} \frac{2}{\pi }{\mathcal {K}}\left( r\right) >\left[ 1-\lambda +\lambda \left( \frac{\text{ arth }r}{r}\right) ^{q}\right] ^{1/q} \end{aligned}$$ holds for $$r\in \left( 0,1\right) $$ with the best constants $$\lambda =3/4$$ and $$q=1/10$$ . This improves some known results and gives a positive answer for a conjecture on the best upper bound for the Gaussian arithmetic–geometric mean in terms of logarithmic and arithmetic means.

The Bhatia—Li mean $$\mathcal {B}_{p}\left( x,y\right) $$ of positive numbers x and y is defined as $$\begin{aligned} \frac{1}{\mathcal {B}_{p}\left( x,y\right) }=\frac{p}{B\left( 1/p,1/p\right) } \int _{0}^{\infty }\frac{dt}{\left( t^{p}+x^{p}\right) ^{1/p}\left( t^{p}+y^{p}\right) ^{1/p}}\text {, }\ p\in \left( 0,\infty \right) , \end{aligned}$$ where $$B\left( \cdot ,\cdot \right) $$ is the Beta function. This new family of means includes the famous logarithmic mean, the Gaussian arithmetic-geometric mean etc. In 2012, Bhatia and Li conjectured that $$\mathcal {B}_{p}\left( x,y\right) $$ is an increasing function of the parameter p on $$\left[ 0,\infty \right] $$ . In this paper, we give a positive answer to this conjecture. Moreover, the mean $$\mathcal {B} _{p}\left( x,y\right) $$ is generalized to an multivariate mean and its elementary properties are investigated.

In this paper, we introduce the notion of n-tuple Diamond-Alpha integral for a function of n variables, and present Jensen, Holder, Minkowski, Beckenbach, and Lyapunov type inequalities for the n-tuple Diamond-Alpha integral on time scales.

Elliptic integrals are of cardinal importance in mathematical analysis and in the field of applied mathematics. Since they cannot be represented by the elementary transcendental functions, there is a need for sharp computable bounds for the family of integrals. In this paper, by studying the monotonicity of the functions $$\begin{aligned} \mathcal {G}_{p}\left( r\right) =\frac{\left( p+r^{2}\right) \mathcal {K} _{a}\left( r\right) }{\ln \left( e^{R\left( a\right) /2}/r^{\prime }\right) } \quad \text {and }\quad \mathcal {I}_{p}\left( r\right) =\frac{\left( p+r^{2}\right) \mathcal {K}_{a}\left( r\right) -p\pi /2}{\ln \left( 1/r^{\prime }\right) } \end{aligned}$$ on $$\left( 0,1\right) $$ , we establish some new sharp lower and upper bounds for the generalized elliptic integrals of the first kind $$\mathcal {K} _{a}\left( r\right) $$ , where $$R\left( x\right) \equiv R\left( x,1-x\right) $$ is the Ramanujan constant function defined on (0, 1 / 2], $$r^{\prime }=\sqrt{ 1-r^{2}}$$ , $$p\in \mathbb {R}$$ is a parameter. These results not only improve some known bounds in the literature, but also yield some new inequalities for $$\mathcal {K}_{a}\left( r\right) $$ .

In this paper, we present a general comparison theorem for two divided differences of a three times differentiable function. This gives a unified treatment for (logarithmically) complete monotonicity, monotonicity and inequalities involving some special functions including gamma, psi and polygamma functions. As their consequences, we not only refine and generalize some important results, but also present simple and interesting alternative proofs of certain earlier results.

In this paper we develop Windschitl’s approximation formula for the gamma function by giving two asymptotic expansions using a little known power series. In particular, for n ∈ ℕ with n ≥ 4, we have Γ x + 1 = 2 π x x e x x sinh ⁡ 1 x x / 2 exp ∑ k = 3 n − 1 a n x 2 n − 1 + R n x $$\begin{array}{} \displaystyle {\it \Gamma} \left( x+1\right) =\sqrt{2\pi x}\left( \tfrac{x}{e}\right) ^{x}\left( x\sinh \tfrac{1}{x}\right) ^{x/2}\exp \left( \sum_{k=3}^{n-1} {\frac{a_{n}}{x^{2n-1}}}+R_{n}\left( x\right) \right) \end{array}$$ with R n x ≤ B 2 n 2 n 2 n − 1 1 x 2 n − 1 $$\begin{array}{} \displaystyle \left\vert R_{n}\left( x\right) \right\vert \leq \frac{\left\vert B_{2n}\right\vert }{2n\left( 2n-1\right) }\frac{1}{x^{2n-1}} \end{array}$$ for all x > 0, where an has a closed-form expression, B 2n is the Bernoulli number. Moreover, we present some approximation formulas for the gamma function related to Windschitl’s approximation, which have higher accuracy.

Hölder’s inequality and its various extensions are playing very important roles in many branches of modern mathematics and physics. In this paper, we present some extensions of Hölder’s inequality via pseudo-integral. Moreover, some related Hölder type inequalities are also given.