Search

Abstract In continuation of Zayed and Bulboacă work in (J. Inequal. Appl. 2022:158, 2022), this paper discusses the geometric characterization of the normalized form of the generalized Bessel function defined by V ρ , r ( z ) : = z + ∑ k = 1 ∞ ( − r ) k 4 k ( 1 ) k ( ρ ) k z k + 1 , z ∈ U , $$\begin{aligned} \mathrm{V}_{\rho,r}(z):=z+\sum_{k=1}^{\infty} \frac{(-r)^{k}}{4^{k}(1)_{k}(\rho )_{k}}z^{k+1}, \quad z\in \mathbb{U}, \end{aligned}$$ for ρ , r ∈ C ∗ : = C ∖ { 0 } $\rho, r\in \mathbb{C}^{\ast}:=\mathbb{C}\setminus \{0\}$ . Precisely, we will use a sharp estimate for the Pochhammer symbol, that is, Γ ( a + n ) / Γ ( a + 1 ) > ( a + α ) n − 1 $\Gamma (a+n)/\Gamma (a+1)>(a+\alpha )^{n-1}$ , or equivalently ( a ) n > a ( a + α ) n − 1 $(a)_{n}>a(a+\alpha )^{n-1}$ , that was firstly proved by Baricz and Ponnusamy for n ∈ N ∖ { 1 , 2 } $n\in \mathbb{N}\setminus \{1,2\}$ , a > 0 $a>0$ and α ∈ [ 0 , 1.302775637 … ] $\alpha \in [0,1.302775637\ldots ]$ in (Integral Transforms Spec. Funct. 21(9):641–653, 2010), and then proved in our paper by another method to improve it using the partial derivatives and the two-variable functions’ extremum technique for n ∈ N ∖ { 1 , 2 } $n\in \mathbb{N}\setminus \{1,2\}$ , a > 0 $a>0$ and 0 ≤ α ≤ 2 $0\leq \alpha \leq \sqrt{2}$ , and used to investigate the orders of starlikeness and convexity. We provide the reader with some examples to illustrate the efficiency of our theory. Our results improve, complement, and generalize some well-known (nonsharp) estimates, as seen in the Concluding Remarks and Outlook section.

Abstract Integral inequalities with generalized convexity play an important role in both applied and theoretical mathematics. The theory of integral inequalities is currently one of the most rapidly developing areas of mathematics due to its wide range of applications. In this paper, we study the concept of higher-order strongly exponentially convex functions and establish a new Hermite–Hadamard inequality for the class of strongly exponentially convex functions of higher order. Further, we derive some new integral inequalities for Riemann–Liouville fractional integrals via higher-order strongly exponentially convex functions. These findings include several well-known results and newly obtained results as special cases. We believe that the results presented in this paper are novel and will be beneficial in encouraging future research in this field.

Abstract In this paper, we first obtain an identity for differentiable mappings. Then, we establish some new generalized inequalities for differentiable convex functions involving some parameters and generalized fractional integrals. We show that these results reduce to several new Simpson-, midpoint- and trapezoid-type inequalities. The results given in this study are the generalizations of results proved in several earlier papers.

Abstract In this paper, we prove an equality for twice-differentiable convex functions involving the conformable fractional integrals. Moreover, several Bullen-type inequalities are established for twice-differentiable functions. More precisely, conformable fractional integrals are used to derive such inequalities. Furthermore, sundry significant inequalities are obtained by taking advantage of the convexity, Hölder inequality, and power-mean inequality. Finally, we provide our results by using special cases of obtained theorems.

Abstract In this paper, we derive a new extension of Hermite-Hadamard’s inequality via k-Riemann-Liouville fractional integrals. Two new k-fractional integral identities are also derived. Then, using these identities as an auxiliary result, we obtain some new k-fractional bounds which involve k-Appell’s hypergeometric functions. These bounds can be viewed as new k-fractional estimations of trapezoidal and mid-point type inequalities. These results are obtained for the functions which have the harmonic convexity property. We also discuss some special cases which can be deduced from the main results of the paper.

Abstract In the paper, we use Jensen’s inequality for diamond integrals and generalize it for n-convex functions with the help of an extended Montgomery identity. Moreover, the bounds have been suggested for identities associated with the generalized Jensen-type functional.

Abstract In this paper we are presenting the refinements of integral inequalities established for convex functions. Consequently, we get refinements of several fractional integral inequalities for different kinds of fractional integral operators.

Abstract The aim of the present paper is to study radii problems for two general classes including various known subclasses of analytic functions associated with the normalized form of Legendre polynomials of odd degree. We also obtain some special cases of the main results presented here with some useful examples.

Abstract The aim of this paper is to introduce a new extension of convexity called σ-convexity. We show that the class of σ-convex functions includes several other classes of convex functions. Some new integral inequalities of Hermite–Hadamard type are established to illustrate the applications of σ-convex functions.

Abstract In this paper, we introduce the notion of exponentially p-convex function and exponentially s-convex function in the second sense. We establish several Hermite–Hadamard type inequalities for exponentially p-convex functions and exponentially s-convex functions in second sense. The present investigation is an extension of several well known results.

Abstract In this paper, we obtain the Hermite–Hadamard type inequalities for s-convex functions and m-convex functions via a generalized fractional integral, known as Katugampola fractional integral, which is the generalization of Riemann–Liouville fractional integral and Hadamard fractional integral. We show that through the Katugampola fractional integral we can find a Hermite–Hadamard inequality via the Riemann–Liouville fractional integral.