Search

In the paper, the author discusses and computes bounds of the sine and cosine along straight lines on the complex plane.

In the paper, by means of the Faa di Bruno formula, with the help of explicit formulas for special values of the Bell polynomials of the second kind with respect to a specific sequence, and by virtue of two combinatorial identities containing the Stirling numbers of the first kind, the author establishes Maclaurin's series expansions for real powers of the inverse cosine (sine) function and the inverse hyperbolic cosine (sine) function. By applying different series expansions for the square of the inverse cosine function, the author not only finds infinite series representations of the circular constant Pi and its square, but also derives two combinatorial identities involving central binomial coefficients.

In the paper, by means of the Faa di Bruno formula, with the help of explicit formulas for special values of the Bell polynomials of the second kind with respect to a specific sequence, and by virtue of two combinatorial identities containing the Stirling numbers of the first kind, the author establishes Maclaurin's series expansions for real powers of the inverse cosine function and the inverse hyperbolic cosine function. By applying different series expansions for the square of the inverse cosine function, the author not only finds infinite series representations of the circular constant Pi and its square, but also derives two combinatorial identities involving central binomial coefficients.

In the paper, the authors apply Faà di Bruno formula, some properties of the Bell polynomials of the second kind, the inversion formulas of binomial numbers and the Stirling numbers of the first and the second kind, to significantly simplify coefficients in two families of ordinary differential equations associated with the higher order Frobenius–Euler numbers.

In the paper, by convolution theorem for the Laplace transforms, some properties of a function involving exponential function, and other analytic techniques, the author finds necessary and sufficient conditions for two functions defined by two derivatives of a function involving trigamma function to be completely monotonic or monotonic.

In the paper, the author finds bounds of the sine and cosine along a circle on the complex plane in terms of two double inequalities for the norms of the sine and cosine along a circle on the complex plane.

In the paper, the authors present an explicit form for a family of inhomogeneous linear ordinary differential equations, find a more significant expression for all derivatives of a function related to the solution to the family of inhomogeneous linear ordinary differential equations in terms of the Lerch transcendent, establish an explicit formula for computing all derivatives of the solution to the family of inhomogeneous linear ordinary differential equations, acquire the absolute monotonicity, complete monotonicity, the Bernstein function property of several functions related to the solution to the family of inhomogeneous linear ordinary differential equations, discover a diagonal recurrence relation of the Stirling numbers of the first kind, and derive an inequality for bounding the logarithmic function.

In this paper, we present derivatives of the generating functions for the Bell numbers by induction and by the Faa di Bruno formula, recover an explicit formula in terms of the Stirling numbers of the second kind, find the (logarithmically) absolute and complete monotonicity of the generating functions, and construct some inequalities for the Bell numbers. From these inequalities, we derive the logarithmic convexity of the sequence of the Bell numbers.

In the paper, the authors establish a new integral identity. By this integral identity and Holder’s inequality, the authors obtain some new inequalities of the Hermite-Hadamard type for e-convex functions.

In the paper, the authors establish explicit formulas for asymptotic and power series expansions of the exponential and the logarithm of asymptotic and power series expansions. The explicit formulas for the power series expansions of the exponential and the logarithm of a power series expansion are applied to find explicit formulas for the Bell numbers and logarithmic polynomials in combinatorics and number theory.

In the paper, by the Cauchy integral formula in the theory of complex functions, an integral representation for the reciprocal of the geometric mean of many positive numbers is established. As a result, the reciprocal of the geometric mean of many positive numbers is verified to be a Stieltjes transform and, consequently, a (logarithmically) completely monotonic function.

In the paper, by the Cauchy integral formula in the theory of complex functions, the authors establish an integral representation for the reciprocal of the weighted geometric mean of two positive numbers. By the integral representation, the authors derive that the reciprocal of the weighted geometric mean of two positive numbers is a Stieltjes function and consequently a (logarithmically) completely monotonic function, present several integral formulas for some kinds of improper integrals, deduce several integral representations for some functions related to the weighted geometric mean, find an equivalent relation between integral representations for the weighted geometric mean and its reciprocal, connect the integral formulas with the central Delannoy numbers, and apply some integral representations and some integral formulas to compute some concrete improper integrals appeared in the theory of complex functions.

In the paper, the authors present by several approaches that both the harmonic mean and the geometric mean of two positive numbers are Bernstein functions and establish their integral representations.

In the paper, the authors introduce the concept “m-AH convex functions” and establish some inequalities of Hermite-Hadamard type for m-AH convex functions.

In the paper, the authors introduce the notion "(α, m; P)-convex function on co-ordinates" and establish new integral inequalities of Hermite-Hadamard type for (α, m; P)-convex functions on co-ordinates in a rectangle from the plane R0×R.

The convexity of functions is a basic concept in mathematics and it has been generalized in various directions. Establishing integral inequalities of Hermite – Hadamard type for various convex functions is one of main topics in the theory of convex functions and attracts a number of mathematicians for several centuries. Currently there have accumulated an amount of literature on integral inequalities of Hermite – Hadamard type for various convex functions. In the paper, the authors introduce a new concept “((α, m), log)–convex functions on the co–ordinates on the rectangle of the plane” and establish new integral inequalities of the Hermite – Hadamard type for ((α, m), log)-convex functions on the co–ordinates on the rectangle of the plane.

In the paper, the authors present a criterion, free of considering differentiability and any partial derivative, to justify a holomorphic function and illustrate the criterion by several examples.

In the paper, the authors introduce a new concept of geometrically r-convex functions and establish some inequalities of Hermite-Hadamard type for this class of functions.

In the paper, the authors establish some integral inequalities of Hermite-Hadamard type involving the product of an s -convex function and a symmetric function and apply these new established inequalities to construct inequalities for special means.

In the paper, the authors confirm the increasing monotonicity of a sequence which originates from computation of the probability of intersecting between a plane couple and a convex body.

In the paper, Schur-geometric and Schur-harmonic convexity of an integral mean for convex functions are established.

In the paper, by virtue of the Faà di Bruno formula, some properties of the Bell polynomials of the second kind, and an inversion formula for the Stirling numbers of the first and second kinds, the authors establish meaningfully and significantly two identities which simplify coefficients in a family of ordinary differential equations associated with higher order Bernoulli numbers of the second kind.

In the note, the authors find several simple and explicit forms for a family of inhomogeneous linear ordinary differential equations studied in "D. Lim, Differential equations for Daehee polynomials and their applications, J. Nonlinear Sci. Appl. 10 (2017), no. 4, 1303-1315; Available online at http://dx.doi.org/10.22436/jnsa.010.04.02".

In the paper, by virtue of techniques in combinatorial analysis, the authors simplify two families of nonlinear ordinary differential equations in terms of the Stirling numbers of the first kind.

In the paper, the authors significantly and meaningfully simplify two families of nonlinear ordinary differential equations in terms of the Stirling numbers of the first and second kinds. Keywords: Nonlinear ordinary differential equation, Stirling number of the first kind, Stirling number of the second kind, Bernoulli number, Euler number

In the paper, by virtue of the Fa\'a di Bruno formula and identities for the Bell polynomials of the second kind, the author simplifies coefficients in a family of ordinary differential equations related to generating functions of reverse Bessel and partially degenerate Bell polynomials.

In the paper, we introduce a new concept ‘geometrically quasi-convex function’ and establish some Hermite–Hadamard type inequalities for functions whose derivatives are of geometric quasi-convexity.

In the paper, the authors obtain some Hermite–Hadamard type integral inequalities for extended s-convex functions on the co-ordinates in a rectangle.

In the paper, the authors introduce a new concept of $r$-$\varphi$-preinvex functions and establish some new integral inequalities of Hermite-Hadamard type.

In the paper, via Riemann-Liouville fractional integration, the authors present some new inequalities of Hermite-Hadamard type for functions whose derivatives in absolute value are (m, h1, h2)-convex.

In the paper, several properties of geometric-arithmetically s-convex functions are provided, an integral identity in which the inte- grands are products of a function and a derivative is found, and then some inequalities of Hermite-Hadamard type for integrals whose inte- grands are products of a derivative and a function whose derivative is of the geometric-arithmetic s-convexity are established.

In the paper, the authors establish some new Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals of functions whose derivatives in absolute value are of (α ,m) -convexity.

In the paper, the authors introduce a new concept “extended $ ((s_1,m_1) $-$ (s_2,m_2)) $-convex function on co-ordinates” and establish some new inequalities of the Hermite–Hadamard type for extended $ ((s_1,m_1) $-$ (s_2,m_2)) $-convex functions of 2 variables on co-ordinates.

In the paper, the authors present an explicit form for a family of inhomogeneous linear ordinary differential equations, find a more significant expression for all derivatives of a function related to the solution to the family of inhomogeneous linear ordinary differential equations in terms of the Lerch transcendent, establish an explicit formula for computing all derivatives of the solution to the family of inhomogeneous linear ordinary differential equations, acquire the absolute monotonicity, complete monotonicity, the Bernstein function property of several functions related to the solution to the family of inhomogeneous linear ordinary differential equations, discover a diagonal recurrence relation of the Stirling numbers of the first kind, and derive an inequality for bounding the logarithmic function.

In the paper, the authors view some ordinary differential equations and their solutions from the angle of (the generalized) derivative polynomials and simplify some known identities for the Bernoulli numbers and polynomials, the Frobenius-Euler polynomials, the Euler numbers and polynomials, in terms of the Stirling numbers of the first and second kinds.

In the paper, the authors generalize the notion “k-Mittag-Leffler function”, establish some integral transforms of the generalized k-Mittag-Leffler function, and derive several special and known conclusions in terms of the generalized Wright function and the generalized k-Wright function. 1. Preliminaries Throughout this paper, let C, R, R0 , R, Z − 0 , and N denote respectively the sets of complex numbers, real numbers, non-negative numbers, positive numbers, non-positive integers, and positive integers. The Pochhammer symbol (λ)ν can be defined for λ, ν ∈ C by (λ)ν = Γ(λ+ν) Γ(λ) , where Γ(z) = lim n→∞ n!n ∏n k=0(z + k) , z ∈ C \ Z0 is called the classical gamma function and its reciprocal 1 Γ is analytic on the whole complex plane C. See [14, Chapter 5], [16, Section 1], and [24, Section 1.1]. In particular, when ν ∈ {0} ∪ N, the quantity (λ)n = { 1, ν = 0 λ(λ+ 1) · · · (λ+ n− 1), n ∈ N is called the rising factorial. See [17] and closely-related references therein. E-mail addresses: qifeng618@gmail.com, qifeng618@hotmail.com, qifeng618@qq.com, ksnisar1@gmail.com, n.sooppy@psau.edu.sa. 2010 Mathematics Subject Classification. Primary 33E12; Secondary 33C20, 44A20, 44A30, 65R10.

By virtue of Cauchy’s integral formula in the theory of complex functions, the authors establish an integral representation for the weighted geometric mean, apply this newly established integral representation to show that the weighted geometric mean is a complete Bernstein function, and find a new proof of the well-known weighted arithmetic-geometric mean inequality.

In the paper, the authors prove that the generalized sine function sin"p","qx and the generalized hyperbolic sine function sinh"p","qx are respectively geometrically concave and geometrically convex. Consequently, the authors verify a conjecture posed by B. A. Bhayo and M. Vuorinen.

In the paper, the author present an inequality for bounding the harmonic number in terms of the hyperbolic cosine.

In the paper, the authors find necessary and sufficient conditions for a difference between the exponential function αe β/t , α, β > 0, and the trigamma function ψ′(t) to be completely monotonic on (0, ∞). While proving the complete monotonicity, the authors discover some properties related to the first order modified Bessel function of the first kind I 1, including inequalities, monotonicity, unimodality, and convexity.

In the paper, the definition of h-geometrically convex functions is introduced, some properties of h-geometrically convex functions are studied, and several integral inequality for the newly defined functions are established.

Submitted to Math. Inequal. Appl. Abstract. In the paper, by creating an integral identity and using H¨ older's inequality, the au- thors establish some new inequalities of Hermite-Hadamard type for n -times differentiable con- vex functions.

In the paper, some new inequalities of Hermite-Hadamard type for (α,m)-convex functions are obtained.

Based on Numerical Manifold Method (NMM), a novel method is presented for computing Stress Intensity Factor (SIF) of cracks. The Williams expansions are employed as the analytical solutions in the mesh containing the crack tip. And high-order polynomial functions are used as the numerical solutions in the surrounding meshes. Then the SIFs are computed via combination of the numerical solutions and the analytical solutions. Meanwhile, the meshes in NMM need not conform to the physical boundaries including the crack edges, allowing the cracks arbitrarily align within the meshes. The given example shows the validity of the method.

In the present study, we prove the monotonicity of two functions involving the gamma function Γ(x) and relating to the n-dimensional volume of the unit ball 𝔹 n in ℝ n .

Summary: In the paper, the famous Hermite-Hadamard integral inequality for convex functionsis generalized to and refined as ones for n-time differentiable functions which are s-convex in thesecond sense, and some known results are improved. 1 Introduction It is common knowledge in mathematical analysis that a function f: I R !R is saidto be convex on I6= ;iff( x+(1 )y) f(x)+(1 )f(y) (1.1)for all x;y2Iand 2[0;1]; If the inequality (1.1) reverses, then the function fis saidto be concave on I.The most important inequality in the theory of convex functions should be the famousHermite-Hadamard inequality (1.2) below. If fis a convex function on [a;b], then0 1b aZ ba f(t)dt f a+b2 f(a)+f(b)21b aZ ba f(t)dt: (1.2)If fis concave on [a;b], then the inequality (1.2) is reversed.In [11], Hermite-Hadamard inequality (1.2) was refined as follows. If f(x) is differ-entiable on [a;b] such that jf 0 (x)j q for q 1 is convex on [a;b], thenZb f(a)+f(b)21b a af(x)dx 1 b a4 jf 0 (a)j q +jf 0 (b)j