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The celebrated Hermite–Hadamard and Ostrowski type inequalities have been studied extensively since they have been established. We find novel versions of the Hermite–Hadamard and Ostrowski type inequalities for the n-polynomial s-type convex functions in the frame of fractional calculus. Taking into account the new concept, we derive some generalizations that capture novel results under investigation. We present two different general techniques, for the functions whose first and second derivativ...

Integral inequality is an interesting mathematical model due to its wide and significant applications in mathematical analysis and fractional calculus. In the present research article, we obtain new inequalities of Simpson’s integral type based on the - convex and - quasiconvex functions in the second derivative sense. In the last sections, some applications on special functions are provided and shown via two figures to demonstrate the explanation of the readers.

In this paper, we present a general definition of a generalized integral operator which contains as particular cases, many of the well-known, fractional and integer order integrals.

Abstract During the last years several fractional integrals were investigated. Having this idea in mind, in the present article, some new generalized fractional integral inequalities of the trapezoidal type for λ φ –preinvex functions, which are differentiable and twice differentiable, are established. Then, by employing those results, we explore the new estimates on trapezoidal type inequalities for classical integral and Riemann–Liouville fractional integrals, respectively. Finally, we apply o...

Integral inequality plays a critical role in both theoretical and applied mathematics fields. It is clear that inequalities aim to develop different mathematical methods (numerically or analytically) and to dedicate the convergence and stability of the methods. Unfortunately, mathematical methods are useless if the method is not convergent or stable. Thus, there is a present day need for accurate inequalities in proving the existence and uniqueness of the mathematical methods. Convexity play a c...

In this paper, we discuss various estimates to the right-hand (resp. left-hand) side of the Hermite–Hadamard inequality for functions whose absolute values of the second (resp. first) derivatives to positive real powers are log-convex. As an application, we derive certain inequalities involving the q-digamma and q-polygamma functions, respectively. As a consequence, new inequalities for the q-analogue of the harmonic numbers in terms of the q-polygamma functions are derived. Moreover, several in...

On some Hermite-Hadamard type inequalities for functions whose second derivative are convex generalized.

In this paper, we establish some new results related to the left-hand of the Hermite-Hadamard type inequalities for the class of functions whose second derivatives are s\varphiconvex functions.

We aim to investigate the following nonlinear boundary value problems of fractional differential equations: $\begin{aligned} (\mathrm{P}_{\lambda}) \left \{ \textstyle\begin{array}{l} -_{t}D_{1}^{\alpha} ( \vert {}_{0}D_{t}^{\alpha}(u(t)) \vert ^{p-2} {}_{0}D_{t}^{\alpha}u(t) ) \\ \quad=f(t,u(t))+\lambda g(t) \vert u(t) \vert ^{q-2}u(t)\quad (t\in(0,1)),\\ u(0)=u(1)=0, \end{array}\displaystyle \right . \end{aligned} where λ is a positive parameter, \(2< r< p< q\), \(\frac{1}{2}<\alpha < 1\), ...

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