In the paper, with aid of the Maclaurin power series expansions of the hyperbolic sine and cosine functions, and in light of stratification method invented and applied by Malesevic and his coauthors, the authors establish several inequalities for bounding the sums of the hyperbolic functions sinhc(2x), tanhc x, and their reciprocals. These inequalities are extensions of the Huygens-type inequalities.

In the paper, the authors establish several new integral inequalities of the Hermite-Hadamard type for functions whose third derivatives are (a, s, m)-convex.

In the paper, the authors discover necessary and sufficient conditions such that an inequality for the ratio between the differences of powers of the tanc function and the cosine function is sound. As a result, the authors derive some new inequalities involving the tangent function, the inverse tangent function, and the second Seiffert mean.

In the paper, the authors establish several identities and relations involving q -analogues of the Pochhammer k -symbol. Moreover, the authors generalize several identities and relations for q -analogues of the Catalan numbers and the Catalan–Qi numbers.

In the paper, in view of two monotonicity rules for the ratios of two functions and of two Maclaurin power series expansions, the authors establish several sharp inequalities involving (hyperbolic) tangent, tanc, cosine, and their reciprocals.

In the study, the authors introduce Qi’s normalized remainder of the Maclaurin power series expansion of the function lnsecx=−lncosx; in view of a monotonicity rule for the ratio of two Maclaurin power series and by virtue of the logarithmic convexity of the function (2x−1)ζ(x) on (1,∞), they prove the logarithmic convexity of Qi’s normalized remainder; with the aid of a monotonicity rule for the ratio of two Maclaurin power series, the authors present the monotonic property of the ratio between two Qi’s normalized remainders.

In the paper, the authors present several inequalities for bounding the sums of two sine cardinal functions and the sums of two hyperbolic sine cardinal functions. These inequalities improve previously-known results.

In this article, the authors introduce Qi’s normalized remainder of the Maclaurin series expansion of Qi’s normalized remainder for the cosine function. By virtue of a monotonicity rule for the quotient of two series and with the aid of an increasing monotonicity of a sequence involving the quotient of two consecutive non-zero Bernoulli numbers, they prove the logarithmic convexity of Qi’s normalized remainder. In view of a higher order derivative formula for the quotient of two functions, they expand the logarithm of Qi’s normalized remainder into a Maclaurin series whose coefficients are expressed in terms of determinants of a class of specific Hessenberg matrices. In light of a monotonicity rule for the quotient of two series, they present the monotonicity of the ratio between two normalized remainders. Finally, the authors connect two of their main results with the generalized hypergeometric functions.

In the paper, the authors introduce a new concept of s-(6, F)-convex functions and establish several integral inequalities of the Hermite-Hadamard type for s-(6, F)-convex functions.

In the paper, the authors find a sufficient and necessary condition for the power-exponential function 1+1xαx to be a Bernstein function, derive closed-form formulas for the nth derivatives of the power-exponential functions 1+1xαx and (1+x)α/x, and present a closed-form formula of the partial Bell polynomials Bn,k(H0(x),H1(x),⋯,Hn−k(x)) for n≥k≥0, where Hk(x)=∫0∞eu−1−ueuuk−1e−xudu for k≥0 are completely monotonic on (0,∞).

In this paper, the authors define the notion of harmonic-arithmetic extended $ (s_1, m_1) $-$ (s_2, m_2) $ coordinated convex functions, establish a new integral identity, present some new Hermite–Hadamard type integral inequalities for harmonic-arithmetic extended $ (s_1, m_1) $-$ (s_2, m_2) $ coordinated convex functions, and derive some known results.

In this paper, the authors define the notion of harmonic-arithmetic extended (s(1), m(1))-(s(2), m(2)) coordinated convex functions, establish a new integral identity, present some new Hermite-Hadamard type integral inequalities for harmonic-arithmetic extended (s(1), m(1))-(s(2), m(2)) coordinated convex functions, and derive some known results.

In this paper, basing on the generating function for the van der Pol numbers, utilizing the Maclaurin power series expansion and two power series expressions of a function involving the cotangent function, and by virtue of the Wronski formula and a derivative formula for the ratio of two differentiable functions, the authors derive four determinantal expressions for the van der Pol numbers, discover two identities for the Bernoulli numbers and the van der Pol numbers, prove the increasing property and concavity of a function involving the cotangent function, and establish two alternative Maclaurin power series expansions of a function involving the cotangent function. The coefficients of the Maclaurin power series expansions are expressed in terms of specific Hessenberg determinants whose elements contain the Bernoulli numbers and binomial coefficients.

In this paper, the authors provide several sharp upper and lower bounds for the Neuman–Sándor mean in terms of the arithmetic and contra-harmonic means, and present some new sharp inequalities involving hyperbolic sine function and hyperbolic cosine function.

In this paper, the authors propose the notions of (α,s)-geometric-arithmetically convex functions and (α,s,m)-geometric-arithmetically convex functions, while they establish some new integral inequalities of the Hermite–Hadamard type for (α,s)-geometric-arithmetically convex functions and for (α,s,m)-geometric-arithmetically convex functions.

In the paper, the authors find series expansions and identities for positive integer powers of inverse (hyperbolic) sine and tangent, for composite of incomplete gamma function with inverse hyperbolic sine, in terms of the first kind Stirling numbers, apply a newly established series expansion to derive a closed-form formula for specific partial Bell polynomials and to derive a series representation of generalized logsine function, and deduce combinatorial identities involving the first kind Stirling numbers.

Let ρ>0 be a constant, let j≥0 be an integer, and let Γ(z) denote the Euler gamma function. With the aid of the integral representation for the Riemann zeta function ζ(z), by virtue of a monotonicity rule, and by means of some properties of the function 1/e^t-1 and its derivatives, the authors discuss the increasing monotonicity of the function t↦t+ρ+jρζ(t+ρ)/ζ(t), study the absolute convexity and logarithmic convexity of the function t(t+j)ζ(t), and derive the increasing monotonicity and inequalities of some sequences involving the ratios |B_2n+2/B_2n| of the Bernoulli numbers B_2n, where zz denotes the extended binomial coefficient.

In the paper, the authors establish a general inequality for the hyperbolic functions, extend the newly-established inequality to trigonometric functions, obtain some new inequalities involving the inverse sine and inverse hyperbolic sine functions, and apply these inequalities to the Neuman–Sándor mean and the first Seiffert mean.

In the paper, with the aid of a known integral identity, the authors establish some new inequalities, similar to the celebrated Simpson's integral inequality, for differentiable MT-convex functions.