It is well-known that in the class of convex functions the (nonnegative) remainder of the Midpoint Rule of the approximate integration is majorized by the remainder of the Trapezoid Rule. Hence the approximation of the integral of the convex function by the Midpoint Rule is better than the analogous approximation by the Trapezoid Rule. Following this fact we examine remainders of certain quadratures in the classes of convex functions of higher orders. Our main results state that for 3-convex (5-convex, respectively) functions the remainder of the 2-point (3-point, respectively) Gauss quadrature is non-negative and it is not greater than the remainder of the Simpson's Rule (4-point Lobatto quadrature, respectively). We also check the 2-point Radau quadratures for 2-convex functions to demonstrate that similar results fail to hold for convex functions of even orders. We apply Peano Kernel Theorem as a~main tool of our considerations.
Motivated by several conjectures posed in the paper " Completely monotonic degrees for a difference between the logarithmic and psi functions",we confirm in this work some conjectures on completely monotonic degrees of remainders of the asymptotic expansion of the logarithm of the gamma function and the digamma function and we give two bounded for this degrees.
The paper deals about Hardy-type inequalities associated with the following higher order Poincaré inequality: $$ \left( \frac{N-1}{2} \right)^{2(k -l)} := \inf_{ u \in C_{c}^{\infty} \setminus \{0\}} \frac{\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{k} u|^2 \ dv_{\mathbb{H}^{N}}}{\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{l} u|^2 \ dv_{\mathbb{H}^{N}} }\,, $$ where $0 \leq l < k$ are integers and $\mathbb{H}^{N}$ denotes the hyperbolic space. More precisely, we improve the Poincaré inequality associated with the above ratio by showing the existence of $k$ Hardy-type remainder terms. F…
Sharp quadrature formulas for integrals of complex rational functions on circles, real axis and its segments are obtained. We also find sharp quadrature formulas for calculation of $L_2$-norms of rational functions on such sets. Basing on quadrature formulas for rational functions, in particular, for simple partial fractions and polynomials, we derive sharp inequalities for different metrics (Nikol'skii type inequalities).
This is a preprint of 1992 with some updates. We study sections of the exponential function Taylor series. Interesting inequalities for these sections were considered by G.Hardy, Kesava Menon, W. Gautschi, H.Alzer and others. The main aim of this preprint is to investigate new proofs for the main inequality with best constants and its multiple generalizations. Some conjectures are formulated (and some of them were proved recently, see comments of 2016).
We obtain new parametric quadrature formulas with variable nodes for integrals of complex rational functions over circles, segments of the real axis and the real axis itself. Basing on these formulas we derive $(q,p)$-inequalities of Jackson-Nikolskii type for various classes of rational functions, complex polynomials and their logarithmic derivatives (simple partial fractions). It is shown that our $(\infty,2)$- and $(\infty,4)$-inequalities are sharp in a number of main theorems. Our inequalities extend and refine several results obtained earlier by other authors.
We give sharp remainder terms of $L^{p}$ and weighted Hardy and Rellich inequalities on one of most general subclasses of nilpotent Lie groups, namely the class of homogeneous groups. As consequences, we obtain analogues of the generalised classical Hardy and Rellich inequalities and the uncertainty principle on homogeneous groups. We also prove higher order inequalities of Hardy-Rellich type, all with sharp constants. A number of identities are derived including weighted and higher order types.
In this paper, we present series representations of the remainders in the expansions for certain trigonometric and hyperbolic functions. By using the obtained results, we establish some inequalities for trigonometric and hyperbolic functions.
We give some estimates of the remainder terms for several conformally-invariant Sobolev-type inequalities on the Heisenberg group, in analogy with the Euclidean case. By considering the variation of associated functionals, we give a stability of two dual forms: the fractional Sobolev (Folland-Stein) and Hardy-Littlewood-Sobolev inequality, in terms of distance to the submanifold of extremizers. Then we compare their remainder terms to improve the inequalities in another way. We also compare, in the limit case s = Q (or $λ$ = 0), the remainder terms of Beckner-Onofri inequality and its dual Log…
In this paper, new sharp weighted generalizations of Ostrowski and generalized trapezoid type inequalities for the Riemann--Stieltjes integrals are proved. Several related inequalities are deduced and investigated. New Simpson's type inequalities for $\mathcal{RS}$--integral are pointed out. Finally, as application; an error estimation of a general quadrature rule for $\mathcal{RS}$--integral via Ostrowski--generalized trapezoid quadrature formula is given.
In this paper our aim is to deduce some sharp Turán type inequalities for the remainder $q-$exponential functions. Our results are shown to be a generalization of results which were obtained by Alzer \cite{al}.
In this paper, we derive new estimates for the remainder term of the midpoint, trapezoid, and Simpson formulae for functions whose derivatives in absolute value at certain power are (α,m)-convex.
In this paper, we derive new estimates for the remainder term of the midpoint, trapezoid, and Simpson formulae for functions whose derivatives in absolute value at certain power are (α,m)-convex.
In this work, Lp-error estimates of general two and three point quadrature rules for Riemann-Stieltjes integrals are give n. The presented proofs depend on new triangle type inequalities of Riemann-Stieltjes integrals
In the paper, we extend Binet's first formula for the logarithm of the gamma function and investigate some properties, including inequalities, star-shaped and sub-additive properties and the complete monotonicity, of the extended remainder of Binet's first formula for the logarithm of the gamma function and related functions.
In this paper, we derive new estimates for the remainder term of the midpoint, trapezoid, and Simpson formulae for functions whose derivatives in absolute value at certain power are P-functions. Some applications to special means of real numbers are also given.
In this paper, we derive new estimates for the remainder term of the midpoint, trapezoid, and Simpson formulae for functions whose derivatives in absolute value at certain power are quasi-convex. Some applications to special means of real numbers are also given.
In this paper, the authors establish some new estimates for the remainder term of the midpoint, trapezoid, and Simpson formula using functions whose derivatives in absolute value at certain power are s-convex. Some applications to special means of real numbers are provided as well.
In this paper, we derive new estimates for the remainder term of the midpoint, trapezoid, and Simpson formulae for functions whose derivatives in absolute value at certain power are quasi-convex. Some applications to special means of real numbers are also given.
In this paper, we obtain some new integral inequalities like Hermite-Hadamard type for third derivatives absolute value are log-convex. We give some applications to quadrature formula for midpoint error estimate.
In this paper, we derive new estimates for the remainder term of the midpoint, trapezoid, and Simpson formulae for functions whose derivatives in absolute value at certain power are h-convex and we point out the results for some special classes of functions. Some applications to special means of real numbers are also given.
In this work, an expansion of Guessab-Schmeisser two points formula for n-times differentiable functions via Fink type identity is established. Generalization of the main result for harmonic sequence of polynomials is established. Several bounds of the presented results are proved. As applications, some quadrature rules are elaborated and discussed. Error bounds of the presented quadrature rules via Chebyshev-Gruss type inequalities are also provided.
We study the Rellich inequalities in the framework of equalities. We present equalities which imply the Rellich inequalities by dropping remainders. This provides a simple and direct understanding of the Rellich inequalities as well as the nonexistence of nontrivial extremisers.
In this paper, a new identity for convex functions is derived. A consequence of the identity is that we can derive new estimates for the remainder term of the midpoint, trapezoid, and Simpson formulae for functions whose derivatives in absolute value at certain power are convex. Some applications to special means of real numbers are also given.
We give new necessary and sufficient conditions for higher order convex ordering. These results generalize the Levin-Stečkin theorem (1960) on convex ordering. The obtained results can be useful in the study of the Hermite-Hadamard type inequalities and in particular inequalities between the quadrature operators.
We consider a generalized form of certain integral inequalities given by Guessab, Schmeisser and Alomari. The trapezoidal, mid point, Simpson, Newton-Simpson rules are obtained as special cases. Also, inequalities for the generalized Alomari functional in terms of the $n$-th order modulus, $n=\overline{1,4}$, are given and applied to some known quadrature rules.
If a smooth function of one variable has maximum one on the unit interval, and has there $d$ zeroes, then its $(d+1)$-st derivative must be "big". This is one of the simplest examples of what we call "smooth rigidity": certain geometric properties of zero sets of smooth functions $f$ imply explicit lower bounds on the high-order derivatives of $f$. In dimensions greater than one, the powerful one-dimension tools, like Lagrange's remainder formula, and divided finite differences, are not directly applicable. Still, the result above implies, via line sections, rather strong restrictions on zeroe…
The purpose of this investigation is to extend basic equations and inequalities which hold for functions $f$ in a Bernstein space $B_σ^2$ to larger spaces by adding a remainder term which involves the distance of $f$ from $B_σ^2$. First we present a modification of the classical modulation space $M^{2,1}(\mathbb{R})$, the so-called readapted modulation space $M^{2,1}_\text{a}(\mathbb{R})$. Our approach to the latter space and its role in functional analysis is novel. In fact, we establish several chains of inclusion relations between $M^{2,1}_\text{a}(\mathbb{R})$ and the more common Lipschitz…
