Deniz Uçar
In this paper, we derive Taylor's theorem for beta-fractional derivative. We also investigate some new properties of Taylor's theorem and some useful related theorems for this derivative. We extend some recent and classical integral inequalities to this new simple interesting fractional calculus including Steffensen and Hermit-Hadamard inequality.
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We derive Taylor's Formula for conformable fractional derivatives. This is then employed to extend some recent and classical integral inequalities to the conformable fractional calculus, including the inequalities of Steffensen, Chebychev, Hermite-Hadamard, Ostrowski, and Gruss.
In this paper the double-sided Taylor's approximations are studied. A short proof of a well-known theorem on the double-sided Taylor's approximations is introduced. Also, two new theorems are proved regarding the monotonicity of such approximations. Then we present some new applications of the double-sided Taylor's approximations in the theory of analytic inequalities.
In this paper the double-sided Talor's approximations are used to obtain generalisations and improvements of some trigonometric 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).
In this paper, we establish a generalized Taylor expansion of a given function $f$ in the form $\displaystyle{f(x) = \sum_{j=0}^m c_j^{α,ρ}\left(x^ρ-a^ρ\right)^{jα} + e_m(x)}$ \noindent with $m\in \mathbb{N}$, $c_j^{α,ρ}\in \mathbb{R}$, $x>a$ and $0< α\leq 1$. In case $ρ= α= 1$, this expression coincides with the classical Taylor formula. The coefficients $c_j^{α,ρ}$, $j=0,\dots,m$ as well as an estimation of $e_m(x)$ are given in terms of the generalized Caputo-type fractional derivatives. Some applications of these results for approximation of functions and for solving some fractional differ…
H. N. Mhaskar obtained the inequalities for derivative of polynomial approximation with Freud type weight. In this paper, we showed the counterpart of the inequalities with Erdos type weight.
In the paper, the authors establish some new Hermite-Hadamard type inequalities for functions whose first derivatives are of convexity and apply these inequalities to construct inequalities of special means.
In this paper, we established new inequalities of Ostrowski's type for the class of preinvex functions are introduced.
In this paper, we obtain new estimates on generalization of Hermite-Hadamard, Simpson and Ostrowski type inequalities for functions whose second derivatives is $φ$-convex via fractional integrals.
In this paper we establish Hermite-Hadamard type inequalities for mappings whose derivatives are s-convex in the second sense and concave.
In this paper, we established some new inequalities via s-convex and s-concave functions.
Holderian functions have strong non-linearities, which result in singularities in the derivatives. This manuscript presents several fractional-order Taylor expansions of Hölderian functions around points of non- differentiability. These expansions are derived using the concept of a fractional velocity, which can be used to describe the singular behavior of derivatives and non-differentiable functions. Fractional velocity is defined as the limit of the difference quotient of the increment of a function and the difference of its argument raised to a fractional power. Fractional velocity can be u…
Here we define a Caputo like discrete nabla fractional difference and we produce discrete nabla fractional Taylor formulae for the first time. We estimate their remaiders. Then we derive related discrete nabla fractional Opial, Ostrowski, Poincare and Sobolev type inequalities.
In this paper several inequalities of the right-hand side of Hermite-Hadamard's inequality are obtained for the class of functions whose derivatives in absolutely value at certain powers are strongly φ-convex with modulus c>0.
Some new inequalities of Ostrowski type for twice differentiable mappings whose derivatives in absolute value are s-convex in the second sense are given.Applications for special means are also provided.
In this paper, we establish several new inequalities for some twice differantiable mappings. Then, we apply these inequalities to obtain new midpoint, trapezoid and perturbed trapezoid rules. Finally, some applications for special means of real numbers are provided.
We establish some new inequalities of Hermite-Hadamard type for functions whose fourth derivatives absolute values are quasi-convex. Further, we give new identity.Using this new identity, we establish similar inequalities for left-hand side of Hermite-Hadamard result.Also, we present applications to special means.
In this paper, the authors establish a new type integral inequalities for differentiable s-convex functions in the second sense. By the well-known Hölder inequality and power mean inequality, they obtain some integral inequalities related to the s-convex functions and apply these inequalities to special means for positive real numbers.
In this paper, we establish several new inequalities for twice differantiable mappings that are connected with the celebrated Hermite-Hadamard integral inequality. Some applications for special means of real numbers are also provided.
Here we define a Caputo like discrete fractional difference and we compare it to the earlier defined Riemann-Liouville fractional discrete analog. Then we produce discrete fractional Taylor formulae for the first time, and we estimate their remainders. Finally, we derive related discrete fractional Ostrowski, Poincare and Sobolev type inequalities.
In this paper several inequalities of the right-hand side of Hermite-Hadamard inequality are obtained for the class of functions whose derivatives in absolutely value at certain powers are (α,m)-convex.Some applications to special means of positive real numbers are also given.
We prove that if $f:I\subset \Bbb R\to \Bbb R$ is of bounded variation, then the noncentered maximal function $Mf$ is absolutely continuous, and its derivative satisfies the sharp inequality $\|DMf\|_1\le |Df|(I)$. This allows us obtain, under less regularity, versions of classical inequalities involving derivatives.
In this paper we establish some estimates of the right hand side of a Hermite-Hadamard type inequality in which some quasi-convex functions are involved.
Anisotropic Caffarelli-Kohn-Nirenberg inequalities with higher-order fractional derivatives
证明了一族涉及高阶分数阶导数和形如$|x'|^{\theta_1}|x|^{\theta_2}|x_n|^{\theta_3}$权重的各向异性Caffarelli-Kohn-Nirenberg插值不等式。
In the paper, the authors find some new integral inequalities of Hermite-Hadamard type for functions whose derivatives of the $n$-th order are $(α,m)$-convex and deduce some known results. As applications of the newly-established results, the authors also derive some inequalities involving special means of two positive real numbers.
Let $$ T(q)=\sum_{k=1}^\infty d(k) q^k, \quad |q|<1, $$ where $d(k)$ denotes the number of positive divisors of the natural number $k$. We present monotonicity properties of functions defined in terms of $T$. More specifically, we proved that $$ H(q) := T(q)- \frac{\log(1-q)}{\log(q)} $$ is strictly increasing in $ (0,1) $ while $$ F(q) := \frac{1-q}{q} \,H(q) $$ is strictly decreasing in $ (0,1) $. These results are then applied to obtain various inequalities, one of which states that the double-inequality $$ α\,\frac{q}{1-q}+\frac{\log(1-q)}{\log(q)} < T(q)< β\,\frac{q}{1-q}+\frac{\log(1-q)}…
In this paper, we present and prove a new truncated $\mathcal{V}$-fractional Taylor's formula using the truncated $\mathcal{V}$-fractional variation of constants formula. In this sense, we present the truncated $\mathcal{V}$-fractional Taylor's remainder by means of $\mathcal{V}$-fractional integral, essential for analyzing and comparing the error, when approaching functions by polynomials. From these new results, some applications were made involving some inequalities, specifically, we generalize the Cauchy-Schwartz inequality.
In this paper, we deal with the convolution series that are a far reaching generalization of the conventional power series and the power series with the fractional exponents including the Mittag-Leffler type functions. Special attention is given to the most interesting case of the convolution series generated by the Sonine kernels. In this paper, we first formulate and prove the second fundamental theorem for the general fractional integrals and the $n$-fold general sequential fractional derivatives of both the Riemann-Liouville and the Caputo types. These results are then employed for derivat…
New identity similar to an identity of [13] for fractional integrals have been defined. Then making use of this identity, some new Ostrowski type inequalities for Riemann-Liouville fractional integral have been developed. Our results have some relationships with the results of Alomari et. al., proved in [13] [published in. Appl. Math. Lett. 23 (2010) 1071-1076] and the analysis used in the proofs is simple.
In this work, we obtain a Lyapunov-type and a Hartman-Wintner-type inequalities for a linear and a nonlinear fractional differential equation with generalized Hilfer operator subject to Dirichlet-type boundary conditions. We prove existence of positive solutions to a nonlinear fractional boundary value problem. As an application, we obtain a lower bound for the eigenvalues of corresponding equations.
We prove a higher order generalization of Glaeser inequality, according to which one can estimate the first derivative of a function in terms of the function itself, and the Holder constant of its k-th derivative. We apply these inequalities in order to obtain pointwise estimates on the derivative of the (k+alpha)-th root of a function of class C^{k} whose derivative of order k is alpha-Holder continuous. Thanks to such estimates, we prove that the root is not just absolutely continuous, but its derivative has a higher summability exponent. Some examples show that our results are optimal.
In this paper, we obtain some new inequalities of Hermite-Hadamard type and Simpson type for functions whose third derivatives belong to Godunova-Levin class.
We first prove some weighted inequalities for compositions of functions on time scales which are in turn applied to establish some new dynamic Opial-type inequalities in several variables. Some generalizations and applications to partial differential dynamic equations are also considered.
Certain Inequalities for the generalized polar derivative of a polynomial
本文研究了多项式广义极导数的不等式，扩展了已知结果并估计了其最大模。
An analogue of Taylor's formula, which arises by substituting the classical derivative by a divided difference operator of Askey-Wilson type, is developed here. We study the convergence of the associated Taylor series. Our results complement a recent work by Ismail and Stanton. Quite surprisingly, in some cases the Taylor polynomials converge to a function which differs from the original one. We provide explicit expressions for the integral remainder. As application, we obtain some summation formulas for basic hypergeometric series. As far as we know, one of them is new. We conclude by studyin…
In this work, an extension of two-point Ostrowski's formula for $n$-times differentiable functions is proved. A generalization of Taylor formula is deduced. An identity of Fink type for this extension is provided. Error estimates for the considered formulas are also given. Two-point Ostrowski-Gruss type inequalities are pointed out. 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 p…
In the paper, we study inequalities for the best trigonometric approximations and fractional moduli of smoothness involving the Weyl and Liouville-Grünwald derivatives in $L_p$, $0<p<1$. We extend known inequalities to the whole range of parameters of smoothness as well as obtain several new inequalities. As an application, the direct and inverse theorems of approximation theory involving the modulus of smoothness $ω_β(f^{(α)},δ)_p$, where $f^{(α)}$ is a fractional derivative of the function $f$, are derived. A description of the class of functions with the optimal rate of decrease of a fracti…
Maximal inequalities for derivatives of spherical means
通过标准球面均值的导数重新表述Stein关于广义球面平均的极大不等式，并证明在特定维度和指数范围内的L^p有界性。
An elementary approach to integral inequalities involving higher order derivatives
受二阶和四阶微分算子分解工作启发，本文通过初等手段证明了一个涉及高阶导数的一般积分不等式，推广了Hardy改进势和Bessel对概念，并给出了众多可由该方法证明的不等式示例。
In this paper, we establish some Simpson type inequalities for functions whose third derivatives in the absolute value are h-convex and (α,m)-convex, respectively.
In this paper we obtained some new Hadamard-Type inequalities for functions whose derivatives absolute values m-convex. Some applications to special means of real numbers are given.
In this paper, we obtain some new inequalities for functions whose second derivatives' absolute value is s-convex and log-convex. Also, we give some applications for numerical integration.
We prove a more general version of the Gruss inequality by using the recent theory of combined dynamic derivatives on time scales and the more general notions of diamond-alpha derivative and integral. For the particular case when alpha = 1, one gets a delta-integral Gruss inequality on time scales; for alpha = 0 a nabla-integral Gruss inequality. If we further restrict ourselves by fixing the time scale to the real (or integer) numbers, then the standard continuous (discrete) inequalities are obtained.
In this paper, we obtain some Simpson type inequalities for functions whose derivatives in absolute value are m- and (alpha,m)- logarithmically convex functions.
Weighted Bernstein-type inequalities on a trapezoidal domain
本文通过单位正方形参数化构造对称二阶 Sturm-Liouville 算子，在梯形域上建立了带显式常数的加权 L^2 方向 Bernstein 不等式，并加强普通偏导数估计，最后将边界因子与 Siciak 极值函数的 Dini 导数关联。
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.
Inequalities for 1/(1-cos(x)) and its derivatives
本文研究函数 $1/(1-\cos x)$ 的单调性质，并给出其各阶导数在 $x,y,x+y$ 组合下的最佳常数界。
In this paper, we obtain some Simpson type inequalities for functions whose second derivatives absolute value or q-th power of them are Q-class functions. Also we give applications to numerical integration.
In this paper, some new inequalities of the Hermite-Hadamard type for functions whose modulus of the derivatives are convex and applications for special means are given. Finally, some error estimates for the trapezoidal formula are obtained.
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.
By using, among other things, the Fourier analysis techniques on hyperbolic and symmetric spaces, we establish the Hardy-Sobolev-Maz'ya inequalities for higher order derivatives on half spaces. The proof relies on a Hardy-Littlewood-Sobolev inequality on hyperbolic spaces which is of its independent interest. We also give an alternative proof of Benguria, Frank and Loss' work concerning the sharp constant in the Hardy-Sobolev-Maz'ya inequality in the three dimensional upper half space. Finally, we show the sharp constant in the Hardy-Sobolev-Maz'ya inequality for bi-Laplacian in the upper half…
Approximation on hexagonal domains by Taylor-Abel-Poisson means
本文研究了积分度量下关于六边形区域周期的多元函数Fourier级数的Taylor-Abel-Poisson线性求和法的逼近性质，证明了直接和逆定理，并得到了Poisson核高阶径向导数$L_1$-范数的Bernstein型不等式。
In the paper, the authors review several refinements of Young's integral inequality via several mean value theorems, such as Lagrange's and Taylor's mean value theorems of Lagrange's and Cauchy's type remainders, and via several fundamental inequalities, such as Čebyšev's integral inequality, Hermite--Hadamard's type integral inequalities, Hölder's integral inequality, and Jensen's discrete and integral inequalities, in terms of higher order derivatives and their norms, survey several applications of several refinements of Young's integral inequality, and further refine Young's integral inequa…
On inequalities between norms of partial derivatives on convex domains
本文研究凸区域上边界为零的二元凹函数偏导数 $L_p$-范数比值的有界性问题，证明了 $p=1$ 时比值始终有界并给出精确估计，而 $p>1$ 时结论依赖于区域几何。
Inequalities for the derivatives
本文证明了在给定函数含噪值及其一阶导数有界时，无法稳定逼近一阶导数，但若已知高于一阶的分数阶导数先验界则可实现，并提出了最优稳定逼近算法及误差估计。
In this paper we achieve some new Hadamard type inequalities using elementary well known inequalities for functions whose first derivatives absolute values are s-geometrically and geometrically convex. And also we get some applications for special means for positive numbers.
In this paper, we obtain some Simpson type inequalities for functions whose derivatives in absolute value are $φ$-convex.
In this paper, we establish some integral inequalities for functions whose second derivatives in absolute value are (α,m)- convex.
Interpolation of derivatives and ultradifferentiable regularity
本文综述任意有限维中$L^p$范数的导数插值不等式，并利用缺项估计在最小权重假设下系统刻画超可微正则性。
This survey discusses the classical Bernstein and Markov inequalities for the derivatives of polynomials, as well as some of their extensions to general sets.
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