Showing papers in "Journal of Inequalities and Applications in 2018"
TL;DR: Several quadratic transformation inequalities for Gaussian hypergeometric function are presented and the analogs of duplication inequalities for the generalized Grötzsch ring function are found.
Abstract: In the article, we present several quadratic transformation inequalities for Gaussian hypergeometric function and find the analogs of duplication inequalities for the generalized Grotzsch ring function
166 citations
TL;DR: In this paper, the Jensen and Hermite-Hadamard type inequalities for interval-valued functions were introduced and generalized for intervalvalued functions, and the Jensen inequalities generalize some known results.
Abstract: In this paper, we introduce the h-convex concept for interval-valued functions. By using the h-convex concept, we present new Jensen and Hermite–Hadamard type inequalities for interval-valued functions. Our inequalities generalize some known results.
93 citations
TL;DR: A new type λ-Bernstein operators with parameter λ∈[−1,1]$\lambda\in[-1, 1]$ is introduced, a Korovkin type approximation theorem is investigated, a local approximation result is established, and a convergence theorem for the Lipschitz continuous functions is given.
Abstract: In this paper, we introduce a new type λ-Bernstein operators with parameter $\lambda\in[-1,1]$
, we investigate a Korovkin type approximation theorem, establish a local approximation theorem, give a convergence theorem for the Lipschitz continuous functions, we also obtain a Voronovskaja-type asymptotic formula. Finally, we give some graphs and numerical examples to show the convergence of $B_{n,\lambda }(f;x)$
to $f(x)$
, and we see that in some cases the errors are smaller than $B_{n}(f)$
to f.
77 citations
TL;DR: The left Riemann–Liouville fractional Hermite–Hadamard type inequalities and the generalized Hermite-Hadamards type inequalities are established by using Green’s function and Jensen‘s inequality and several new Hermite—HadamARD type inequalities for a class of convex as well as monotone functions are presented.
Abstract: In the article, we establish the left Riemann–Liouville fractional Hermite–Hadamard type inequalities and the generalized Hermite–Hadamard type inequalities by using Green’s function and Jensen’s inequality, and present several new Hermite–Hadamard type inequalities for a class of convex as well as monotone functions.
77 citations
TL;DR: Several Ostrowski type inequalities involving the conformable fractional integrals are established and new inequalities for the arithmetic and generalized logarithmic means are found.
Abstract: In the article, we establish several Ostrowski type inequalities involving the conformable fractional integrals. As applications, we find new inequalities for the arithmetic and generalized logarithmic means.
69 citations
TL;DR: Several monotonicity and convexity properties are generalized as well as sharp inequalities for the complete elliptic integrals to the complete p-ellipticintegrals.
Abstract: We generalize several monotonicity and convexity properties as well as sharp inequalities for the complete elliptic integrals to the complete p-elliptic integrals.
67 citations
TL;DR: New oscillation criteria are established, and they essentially improve the well-known results reported in the literature, including those for non-neutral differential equations, by taking into account such part of the overall impact of the delay that has been neglected in earlier results.
Abstract: The paper is devoted to the study of oscillation of solutions to a class of second-order half-linear neutral differential equations with delayed arguments. New oscillation criteria are established, and they essentially improve the well-known results reported in the literature, including those for non-neutral differential equations. The adopted approach refines the classical Riccati transformation technique by taking into account such part of the overall impact of the delay that has been neglected in the earlier results. The effectiveness of the obtained criteria is illustrated via examples.
58 citations
TL;DR: Two new half-discrete Hilbert inequalities for three variables are obtained with the best constant factor and are given their equivalent forms.
Abstract: In this paper, we obtain two new half-discrete Hilbert inequalities for three variables. The obtained inequalities are with the best constant factor. Moreover, we give their equivalent forms.
48 citations
TL;DR: The Bohr-type radius of the alternating series associated with the Taylor series of analytic functions is investigated and it is proved that most of the results are sharp.
Abstract: In this paper, we investigate the Bohr-type radii for several different forms of Bohr-type inequalities of analytic functions in the unit disk, we also investigate the Bohr-type radius of the alternating series associated with the Taylor series of analytic functions. We will prove that most of the results are sharp.
48 citations
TL;DR: Different type inequalities via quantum integrals are established based on (α, m)$(\alpha,m)$-convexity, which generalize some results given in the literature.
Abstract: In this paper, based on $(\alpha,m)$
-convexity, we establish different type inequalities via quantum integrals. These inequalities generalize some results given in the literature.
47 citations
TL;DR: Several sharp upper and lower bounds for the generalized Euler–Mascheroni constant are provided, as consequences, some previous bounds are improved.
Abstract: We provide several sharp upper and lower bounds for the generalized Euler–Mascheroni constant. As consequences, some previous bounds for the Euler–Mascheroni constant are improved.
TL;DR: The authors discover a general k-fractional integral identity with multi-parameters for twice differentiable functions and derive some new bounds on Hermite–Hadamard’s and Simpson's inequalities for generalized (m,h)$(m, h)-preinvex functions through k- fractional integrals.
Abstract: The authors discover a general k-fractional integral identity with multi-parameters for twice differentiable functions. By using this integral equation, the authors derive some new bounds on Hermite–Hadamard’s and Simpson’s inequalities for generalized $(m,h)$
-preinvex functions through k-fractional integrals. By taking the special parameter values for various suitable choices of function h, some interesting results are also obtained.
TL;DR: In the paper, the authors present some inequalities involving the extended gamma function and the Kummer confluent hypergeometric k-function via some classical inequalities such as Chebychev’s inequality for synchronous and asynchronous mappings, and introduce a Turán type mean inequality for the Kummers confluent k-hypergeometric function.
Abstract: In the paper, the authors present some inequalities involving the extended gamma function and the Kummer confluent hypergeometric k-function via some classical inequalities such as Chebychev’s inequality for synchronous (or asynchronous, respectively) mappings, give a new proof of the log-convexity of the extended gamma function by using the Holder inequality, and introduce a Turan type mean inequality for the Kummer confluent k-hypergeometric function.
TL;DR: A quantitative Voronovskaja type theorem by means of Ditzian–Totik modulus of smoothness is proved and a Grüss–Voronovski type theorem for λ-Kantorovich operators is provided.
Abstract: In the present paper, we study a new type of Bernstein operators depending on the parameter $\lambda\in[-1,1]$
. The Kantorovich modification of these sequences of linear positive operators will be considered. A quantitative Voronovskaja type theorem by means of Ditzian–Totik modulus of smoothness is proved. Also, a Gruss–Voronovskaja type theorem for λ-Kantorovich operators is provided. Some numerical examples which show the relevance of the results are given.
TL;DR: By using the coincidence degree theory, an existence result is presented for a coupled system of nonlinear fractional differential equations with multi-point boundary conditions at resonance.
Abstract: By using the coincidence degree theory, we present an existence result for a coupled system of nonlinear fractional differential equations with multi-point boundary conditions at resonance.
TL;DR: Based upon a fixed point theorem of increasing φ-(h,r)$(h, r)$-concave operators, the existence and uniqueness of solutions for the problem dependent on two constants lf,lg$l_{f}, l_{g}$.
Abstract: In this article, we discuss a new Hadamard fractional differential system with four-point boundary conditions $$\textstyle\begin{cases} {}^{H} D^{\alpha}u(t)+f(t,v(t))=l_{f},\quad t\in(1,e),\\ {}^{H} D^{\beta}v(t)+g(t,u(t))=l_{g},\quad t\in(1,e),\\ u^{(j)}(1)=v^{(j)}(1)=0, \quad 0\leq j\leq n-2,\\ u(e)=av(\xi),\qquad v(e)=bu(\eta),\quad \xi, \eta\in(1,e), \end{cases} $$
where $a,b$
are two parameters with $0< ab(\log\eta)^{\alpha-1}(\log\xi )^{\beta-1}<1$
, $\alpha, \beta\in(n-1,n]$
are two real numbers and $n\geq3$
, $f,g\in C([1,e]\times(-\infty,+\infty),(-\infty,+\infty))$
, $l_{f}, l_{g}>0$
are constants, and ${}^{H} D^{\alpha}, {}^{H} D^{\beta}$
are the Hadamard fractional derivatives of fractional order. Based upon a fixed point theorem of increasing φ-
$(h,r)$
-concave operators, we establish the existence and uniqueness of solutions for the problem dependent on two constants $l_{f}, l_{g}$
.
TL;DR: Through the notion of F-convex, some new Hermite–Hadamard type and trapezoid type inequalities are found for the Riemann–Liouville fractional integrals and classical integrals.
Abstract: In this study, the family F and F-convex function are given with its properties. In view of this, we establish some new inequalities of Hermite-Hadamard type for differentiable function. Moreover, we establish some trapezoid type inequalities for functions whose second derivatives in absolute values are F-convex. We also show that through the notion of F-convex we can find some new Hermite-Hadamard type and trapezoid type inequalities for the Riemann-Liouville fractional integrals and classical integrals.
TL;DR: The Hadamard and the Fejér–Hadamard inequalities for the extended generalized fractional integral operator involving the extended summarized generalized Mittag-Leffler function are proved.
Abstract: In this paper we prove the Hadamard and the Fejer–Hadamard inequalities for the extended generalized fractional integral operator involving the extended generalized Mittag-Leffler function. The extended generalized Mittag-Leffler function includes many known special functions. We have several such inequalities corresponding to special cases of the extended generalized Mittag-Leffler function. Also there we note the known results that can be obtained.
TL;DR: The components of SE$S$ as a pre-quasi Banach operator ideal containing finite dimensional operators as a dense subset and its completeness are proved and it is shown that the operator ideal formed by E$\mathbb{E}$ and approximation numbers is small under certain conditions.
Abstract: In this article, we establish sufficient conditions on the generalized Cesaro and Orlicz sequence spaces $\mathbb{E}$
such that the class $S_{\mathbb{E}}$
of all bounded linear operators between arbitrary Banach spaces with its sequence of s-numbers belonging to $\mathbb{E}$
generates an operator ideal. The components of $S_{\mathbb{E}}$
as a pre-quasi Banach operator ideal containing finite dimensional operators as a dense subset and its completeness are proved. Some inclusion relations between the operator ideals as well as the inclusion relations for their duals are obtained. Finally, we show that the operator ideal formed by $\mathbb{E}$
and approximation numbers is small under certain conditions.
TL;DR: In the article, several sharp upper and lower bounds for two Sándor–Yang means in terms of combinations of arithmetic and contra-harmonic means are provided.
Abstract: In the article, we provide several sharp upper and lower bounds for two Sandor–Yang means in terms of combinations of arithmetic and contra-harmonic means.
TL;DR: It is established that some new noninstantaneous impulsive inequalities using the conformable fractional calculus are established.
Abstract: We establish some new noninstantaneous impulsive inequalities using the conformable fractional calculus.
TL;DR: The rate of convergence of new operators via a Peetre K$\mathcal{K}$-functional and corresponding modulus of smoothness, quantitative Voronovskaya type theorem and Grüss–Voronovo type theorem in quantitative mean are discussed and a graphic for new operators with special cases and for some values of n is presented.
Abstract: The present paper deals with genuine Bernstein–Durrmeyer operators which preserve some certain functions. The rate of convergence of new operators via a Peetre $\mathcal{K}$
-functional and corresponding modulus of smoothness, quantitative Voronovskaya type theorem and Gruss–Voronovskaya type theorem in quantitative mean are discussed. Finally, the graphic for new operators with special cases and for some values of n is also presented.
TL;DR: New generalized Lyapunov-type and Hartman-type inequalities are obtained when the conformable derivative Tαa$\mathbf{T}_{\alpha }^{a}$ is replaced by a sequential conformable derivatives T αa∘Tαa $.
Abstract: We state and prove new generalized Lyapunov-type and Hartman-type inequalities for a conformable boundary value problem of order $\alpha \in (1,2]$
with mixed non-linearities of the form $$ \bigl(\mathbf{T}_{\alpha }^{a} x\bigr) (t)+r_{1}(t) \bigl\vert x(t) \bigr\vert ^{\eta -1}x(t)+r_{2}(t)\bigl\vert x(t) \bigr\vert ^{ \delta -1}x(t)=g(t), \quad t\in (a,b), $$
satisfying the Dirichlet boundary conditions $x(a)=x(b)=0$
, where $r_{1}$
, $r_{2}$
, and g are real-valued integrable functions, and the non-linearities satisfy the conditions $0<\eta <1<\delta <2$
. Moreover, Lyapunov-type and Hartman-type inequalities are obtained when the conformable derivative $\mathbf{T}_{\alpha }^{a}$
is replaced by a sequential conformable derivative $\mathbf{T}_{\alpha }^{a} \circ \mathbf{T}_{\alpha }^{a}$
, $\alpha \in (1/2,1]$
. The potential functions $r_{1}$
, $r_{2}$
as well as the forcing term g require no sign restrictions. The obtained inequalities generalize some existing results in the literature.
TL;DR: The Bézier variant of Kantorovich type λ-Bernstein operators with parameter λ∈[−1,1]$\lambda\in[-1, 1]$.
Abstract: In this paper, we introduce the Bezier variant of Kantorovich type λ-Bernstein operators with parameter $\lambda\in[-1,1]$
. We establish a global approximation theorem in terms of second order modulus of continuity and a direct approximation theorem by means of the Ditzian–Totik modulus of smoothness. Finally, we combine the Bojanic–Cheng decomposition method with some analysis techniques to derive an asymptotic estimate on the rate of convergence for some absolutely continuous functions.
TL;DR: A Lyapunov-type inequality is established for the anti-periodic fractional boundary value problem and an application of the obtained inequality to the corresponding eigenvalue problem is given.
Abstract: A Lyapunov-type inequality is established for the anti-periodic fractional boundary value problem $$\begin{aligned} & \bigl({}^{C}D_{a}^{\alpha,\psi}u \bigr) (x)+f \bigl(x,u(x) \bigr)=0,\quad a< x< b, \\ &u(a)+u(b)=0,\qquad u'(a)+u'(b)=0, \end{aligned}$$ where $(a,b)\in\mathbb{R}^{2}$
, $a< b$
, $1<\alpha<2$
, $\psi\in C^{2}([a,b])$
, $\psi'(x)>0$
, $x\in[a,b]$
, ${}^{C}D_{a}^{\alpha,\psi}$
is the ψ-Caputo fractional derivative of order α, and $f: [a,b]\times\mathbb{R}\to\mathbb{R}$
is a given function. Next, we give an application of the obtained inequality to the corresponding eigenvalue problem.
TL;DR: The existence of common fixed points of a certain mapping in the frame of an extended b-metric space is investigated and a number of well-known fixed point theorems in the literature are covered.
Abstract: In this paper, we investigate the existence of common fixed points of a certain mapping in the frame of an extended b-metric space. The given results cover a number of well-known fixed point theorems in the literature. We state some examples to illustrate our results.
TL;DR: This paper establishes a new Korovkin-type approximation theorem for the functions of two variables defined on a Banach space CB(D)$C_{B}(\mathcal{D})$ and presents an illustrative example to show that this result is a non-trivial extension of some traditional and statistical versions of Korvkin- type approximation theorems which were demonstrated in the earlier works.
Abstract: The notion of statistical weighted $\mathcal{B}$
-summability was introduced very recently (Kadak et al. in Appl. Math. Comput. 302:80–96, 2017). In the paper, we study the concept of statistical deferred weighted $\mathcal{B}$
-summability and deferred weighted $\mathcal{B}$
-statistical convergence and then establish an inclusion relation between them. In particular, based on our proposed methods, we establish a new Korovkin-type approximation theorem for the functions of two variables defined on a Banach space $C_{B}(\mathcal{D})$
and then present an illustrative example to show that our result is a non-trivial extension of some traditional and statistical versions of Korovkin-type approximation theorems which were demonstrated in the earlier works. Furthermore, we establish another result for the rate of deferred weighted $\mathcal{B}$
-statistical convergence for the same set of functions via modulus of continuity. Finally, we consider a number of interesting special cases and illustrative examples in support of our findings of this paper.
TL;DR: From sums of finite products of Chebyshev polynomials of the second kind and of Fibonacci polynmials, the Fourier series expansions of functions associated with them are derived and some identities are obtained by using those expressions.
Abstract: In this paper, we consider sums of finite products of Chebyshev polynomials of the second kind and of Fibonacci polynomials and derive Fourier series expansions of functions associated with them. From these Fourier series expansions, we can express those sums of finite products in terms of Bernoulli polynomials and obtain some identities by using those expressions.
TL;DR: Some approximation results are obtained for these new Dunkl type Szász operators via post-quantum calculus and the rate of convergence of these operators for functions belonging to the Lipschitz class is obtained.
Abstract: The object of this paper to construct Dunkl type Szasz operators via post-quantum calculus. We obtain some approximation results for these new operators and compute convergence of the operators by using the modulus of continuity. Furthermore, we obtain the rate of convergence of these operators for functions belonging to the Lipschitz class. We also study the bivariate version of these operators.
TL;DR: In this paper some new general fractional integral inequalities for convex and m-convex functions by involving an extended Mittag-Leffler function are presented.
Abstract: In this paper some new general fractional integral inequalities for convex and m-convex functions by involving an extended Mittag-Leffler function are presented These results produce inequalities for several kinds of fractional integral operators Some interesting special cases of our main results are also pointed out