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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 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 article, we prove that the double inequalities α1[7C(a,b)16+9H(a,b)16]+(1−α1)[3A(a,b)4+G(a,b)4]0\\(a, b>0\\) with a≠b\\(a b\\) if and only if α1≤3/16=0.1875\\(_1 3/16=0.1875\\), β1≥64/π2−6=0.484555…\\(_164/^2-6= 0.484555\\), α2≤3/16=0.1875\\(_23/16=0.1875\\) and β2≥(5log2−log3−2logπ)/(log7−log6)=0.503817…\\(_2(52-3-2 )/(7-6)= 0.503817\\), where E(a,b)=(2π∫0π/2acos2θ+bsin2θdθ)2\\(E(a,b)= (2ınt^/2_0a^2 +b^2\\,d )^2\\), H(a,b)=2ab/(a+b)\\(H(a,b)=2ab/(a+b)\\), G(a,b)=ab\\(G(a,b)=ab\\), A(a,b)=(a+b)/2\\(A(a,b)=(a+b)/2\\) and C(a,b)=(a2+b2)/(a+b)\\(C(a,b)=(a^2+b^2)/(a+b)\\) are the quasi-arithmetic, harmonic, geometric, arithmetic and contra-harmonic means of a and b, respectively.

Although most of the early research studies on fractional-order systems were based on the Caputo or Riemann–Liouville fractional-order derivatives, it has recently been proven that these methods have some drawbacks. For instance, kernels of these methods have a singularity that occurs at the endpoint of an interval of definition. Thus, to overcome this issue, several new definitions of fractional derivatives have been introduced. The Caputo–Fabrizio fractional order is one of these nonsingular definitions. This paper is concerned with the analyses and design of an optimal control strategy for a Caputo–Fabrizio fractional-order model of the HIV/AIDS epidemic. The Caputo–Fabrizio fractional-order model of HIV/AIDS is considered to prevent the singularity problem, which is a real concern in the modeling of real-world systems and phenomena. Firstly, in order to find out how the population of each compartment can be controlled, sensitivity analyses were conducted. Based on the sensitivity analyses, the most effective agents in disease transmission and prevalence were selected as control inputs. In this way, a modified Caputo–Fabrizio fractional-order model of the HIV/AIDS epidemic is proposed. By changing the contact rate of susceptible and infectious people, the atraumatic restorative treatment rate of the treated compartment individuals, and the sexual habits of susceptible people, optimal control was designed. Lastly, simulation results that demonstrate the appropriate performance of the Caputo–Fabrizio fractional-order model and proposed control scheme are illustrated.

In this work, we introduce the idea of n–polynomial harmonically s–type convex function. We elaborate the new introduced idea by examples and some interesting algebraic properties. As a result, new Hermite–Hadamard, some refinements of Hermite–Hadamard and Ostrowski type integral inequalities are established, which are the generalized variants of the previously known results for harmonically convex functions. Finally, we illustrate the applicability of this new investigation in special functions (hypergeometric function and special mean of real numbers).

For $r\\in(0,1)$, let $\\mu \\left( r\\right) $ be the modulus of the plane Grötzsch ring $\\mathbb{B}^2\\setminus[0,r]$, where $\\mathbb{B}^2$ is the unit disk. In this paper, we prove that \\begin{equation*} \\mu \\left( r\\right) =\\ln \\frac{4}{r}-\\sum_{n=1}^{\\infty }\\frac{\\theta _{n}}{ 2n}r^{2n}, \\end{equation*} with $\\theta _{n}\\in \\left( 0,1\\right)$. Employing this series expansion, we obtain several absolutely monotonic and (logarithmically) completely monotonic functions involving $\\mu \\left( r\\right) $, which yields some new results and extend certain known ones. Moreover, we give an affirmative answer to the conjecture proposed by Alzer and Richards in H. Alzer and K. Richards, On the modulus of the Grötzsch ring, J. Math. Anal. Appl. 432(1): (2015), 134–141, DOI 10.1016/j.jmaa.2015.06.057. As applications, several new sharp bounds and functional inequalities for $\\mu \\left( r\\right) $ are established.

In accordance with the quantum calculus, we introduced the two variable forms of Hermite-Hadamard- (HH-) type inequality over finite rectangles for generalized Ψ-convex functions. This novel framework is the convolution of quantum calculus, convexity, and special functions. Taking into account the q^1q^2-integral identity, we demonstrate the novel generalizations of the HH-type inequality for q^1q^2-differentiable function by acquainting Raina’s functions. Additionally, we present a different approach that can be used to characterize HH-type variants with respect to Raina’s function of coordinated generalized Ψ-convex functions within the quantum techniques. This new study has the ability to generate certain novel bounds and some well-known consequences in the relative literature. As application viewpoint, the proposed study for changing parametric values associated with Raina’s functions exhibits interesting results in order to show the applicability and supremacy of the obtained results. It is expected that this method which is very useful, accurate, and versatile will open a new venue for the real-world phenomena of special relativity and quantum theory.

For r ∈ (0, 1), let 𝒦(r) be the complete elliptic integral of the first kind. In this paper, by introducing two classes of logarithmic-type functions, and proving the monotonicity and absolutely monotonicity properties of certain functions involving 𝒦(r) and the logarithmic-type functions, several new functional inequalities for 𝒦(r) will be derived, which improve some previous known results.

We find the greatest values p1, p2 and least values q1, q2 such that the double inequalities Sp1(a,b)0 with a≠b and present some new bounds for the complete elliptic integrals. Here M(a,b), T(a,b), and Sp(a,b) are the arithmetic-geometric, Toader, and pth Gini means of two positive numbers a and b, respectively.

A new rule for calculating the parameter t involved in each iteration of the MHSDL (Dai-Liao) conjugate gradient (CG) method is presented. The new value of the parameter initiates a more efficient and robust variant of the Dai-Liao algorithm. Under proper conditions, theoretical analysis reveals that the proposed method in conjunction with backtracking line search is of global convergence. Numerical experiments are also presented, which confirm the influence of the new value of the parameter t on the behavior of the underlying CG optimization method. Numerical comparisons and the analysis of obtained results considering Dolan and Moré’s performance profile show better performances of the novel method with respect to all three analyzed characteristics: number of iterative steps, number of function evaluations, and CPU time.