Search Results (22)

In this paper, we establish some Simpson-type inequalities within the framework of Riemann–Liouville fractional calculus, specifically tailored for differentiable harmonically convex functions. By introducing a novel fractional integral identity for differentiable functions with harmonic arguments, we derive several estimates that generalize and refine existing results in the literature. The theoretical findings are validated through a numerical example supported by graphical illustration, and potential applications in approximation theory and numerical analysis are discussed. Full article

In this paper, we investigate Hermite–Hadamard-type inequalities for harmonically s-convex stochastic processes via Riemann–Liouville fractional integrals. We begin by introducing the notion of harmonically s-convex stochastic processes. Subsequently, we establish a variety of Riemann–Liouville fractional Hermite–Hadamard-type inequalities for harmonic s-convex stochastic. We first provide the Hermite–Hadamard inequality, then by introducing a novel identity involving mean-square stochastic Riemann–Liouville fractional integral operators, we derive several midpoint-type inequalities for harmonically s-convex stochastic processes. Illustrative example with graphical depiction and a practical application are provided. Full article

Here in this paper, we establish the basic inclusion relations among the harmonic class $\mathbb{HF}(\sigma ,\eta )$ with the classes $S_{\mathcal{HF}}^{*}$ of starlike harmonic functions and $K_{\mathcal{HF}}$ of convex harmonic functions defined in open symmetric unit disk U. Moreover, we investigate inclusion connections for the harmonic classes $\mathcal{T}{\mathfrak{N}}_{\mathbb{HF}}\left(\varrho \right)$ and $\mathcal{T}{\mathfrak{Q}}_{\mathbb{HF}}\left(\varrho \right)$ of harmonic functions by applying the operator $\mathrm{\Lambda }$ associated with the Bessel function. Furthermore, several special cases of the main results are obtained for the particular case $\sigma =0$. Full article

In this paper, we aim to establish new inequalities of Hermite–Hadamard (H.H) type for harmonically convex functions using proportional Caputo-Hybrid (P.C.H) fractional operators. Parameterized by $\alpha$, these operators offer a unique flexibility: setting $\alpha =1$ recovers the classical inequalities for harmonically convex functions, while setting $\alpha =0$ yields inequalities for differentiable harmonically convex functions. This framework allows us to unify classical and fractional cases within a single operator. To validate the theoretical results, we provide several illustrative examples supported by graphical representations, marking the first use of such visualizations for inequalities derived via P.C.H operators. Additionally, we demonstrate practical applications of the results by deriving new fractional-order recurrence relations for the modified Bessel function of type-1, which are useful in mathematical modeling, engineering, and physics. The findings contribute to the growing body of research in fractional inequalities and harmonic convexity, paving the way for further exploration of generalized convexities and higher-order fractional operators. Full article

Cancer cells typically divide with weaker synchronisation with the circadian clock than normal cells, with the degree of decoupling increasing with tumour maturity. Chronotherapy exploits this loss of synchronisation, using drugs with circadian-clock-dependent activity and timed infusion to balance the competing demands of reducing toxicity toward normal cells that display physiological circadian rhythms and of efficacy against the tumour. We analysed optimal chronotherapy for one-compartment nonlinear tumour growth models that were no longer synchronised with the circadian clock, minimising a cost function with a periodically driven running cost accounting for the circadian drug tolerability of normal cells. Using Pontryagin’s Minimum Principle (PMP), we show, for drugs that either increase the cell death rate or kill dividing cells, that optimal solutions are aperiodic bang–bang solutions with two switches per day, with the duration of the daily drug administration increasing as treatment progresses; for large tumours, optimal therapy can in fact switch mid treatment from aperiodic to continuous treatment. We illustrate this with tumours grown under logistic and Gompertz dynamics conditions; for logistic growth, we categorise the different types of solutions. Singular solutions can be applicable for some nonlinear tumour growth models if the per capita growth rate is convex. Direct comparison of the optimal aperiodic solution with the optimal periodic solution shows the former presents reduced toxicity whilst retaining similar efficacy against the tumour. We only found periodic solutions with a daily period in one-compartment exponential growth models, whilst models incorporating nonlinear growth had generic aperiodic solutions, and linear multi-compartments appeared to have long-period (weeks) periodic solutions. Our results suggest that chronotherapy-based optimal solutions under a harmonic running cost are not typically periodic infusion schedules with a 24 h period. Full article

This paper presents an extensive review of some recent results on fractional Ostrowski-type inequalities associated with a variety of convexities and different kinds of fractional integrals. We have taken into account the classical convex functions, quasi-convex functions, $(\zeta ,m)$-convex functions, s-convex functions, $(s,r)$-convex functions, strongly convex functions, harmonically convex functions, h-convex functions, Godunova-Levin-convex functions, $MT$-convex functions, P-convex functions, m-convex functions, $(s,m)$-convex functions, exponentially s-convex functions, $(\beta ,m)$-convex functions, exponential-convex functions, $\overline{\zeta },\beta ,\gamma ,\delta$-convex functions, quasi-geometrically convex functions, $s-e$-convex functions and n-polynomial exponentially s-convex functions. Riemann–Liouville fractional integral, Katugampola fractional integral, k-Riemann–Liouville, Riemann–Liouville fractional integrals with respect to another function, Hadamard fractional integral, fractional integrals with exponential kernel and Atagana-Baleanu fractional integrals are included. Results for Ostrowski-Mercer-type inequalities, Ostrowski-type inequalities for preinvex functions, Ostrowski-type inequalities for Quantum-Calculus and Ostrowski-type inequalities of tensorial type are also presented. Full article

A review of the results on the fractional Fejér-type inequalities, associated with different families of convexities and different kinds of fractional integrals, is presented. In the numerous families of convexities, it includes classical convex functions, s-convex functions, quasi-convex functions, strongly convex functions, harmonically convex functions, harmonically quasi-convex functions, quasi-geometrically convex functions, p-convex functions, convexity with respect to strictly monotone function, co-ordinated-convex functions, $(\theta ,h-m)-p$-convex functions, and h-preinvex functions. Included in the fractional integral operators are Riemann–Liouville fractional integral, $(k-p)$-Riemann–Liouville, k-Riemann–Liouville fractional integral, Riemann–Liouville fractional integrals with respect to another function, the weighted fractional integrals of a function with respect to another function, fractional integral operators with the exponential kernel, Hadamard fractional integral, Raina fractional integral operator, conformable integrals, non-conformable fractional integral, and Katugampola fractional integral. Finally, Fejér-type fractional integral inequalities for invex functions and $(p,q)$-calculus are also included. Full article

The term convexity and theory of inequalities is an enormous and intriguing domain of research in the realm of mathematical comprehension. Due to its applications in multiple areas of science, the theory of convexity and inequalities have recently attracted a lot of attention from historians and modern researchers. This article explores the concept of a new group of modified harmonic exponential s-convex functions. Some of its significant algebraic properties are elegantly elaborated to maintain the newly described idea. A new sort of Hermite–Hadamard-type integral inequality using this new concept of the function is investigated. In addition, several new estimates of Hermite–Hadamard inequality are presented to improve the study. These new results illustrate some generalizations of prior findings in the literature. Full article

In this study, we consider different types of convex-exponent products of elements of a certain class of log-harmonic mapping and then find sufficient conditions for them to be starlike log-harmonic functions. For instance, we show that, if f is a spirallike function, then choosing a suitable value of $\gamma$, the log-harmonic mapping $F\left(z\right)={f\left(z\right)\left|f\left(z\right)\right|}^{2\gamma }$ is $\alpha$-$spiralike$ of order $\rho$. Our results generalize earlier work in the literature. Full article

In the frame of fractional calculus, the term convexity is primarily utilized to address several challenges in both pure and applied research. The main focus and objective of this review paper is to present Hermite–Hadamard (H-H)-type inequalities involving a variety of classes of convexities pertaining to fractional integral operators. Included in the various classes of convexities are classical convex functions, m-convex functions, r-convex functions, $(\alpha ,m)$-convex functions, $(\alpha ,m)$-geometrically convex functions, harmonically convex functions, harmonically symmetric functions, harmonically $(\theta ,m)$-convex functions, m-harmonic harmonically convex functions, $(s,r)$-convex functions, arithmetic–geometric convex functions, logarithmically convex functions, $(\alpha ,m)$-logarithmically convex functions, geometric–arithmetically s-convex functions, s-convex functions, Godunova–Levin-convex functions, differentiable $\varphi$-convex functions, $MT$-convex functions, $(s,m)$-convex functions, p-convex functions, h-convex functions, $\sigma$-convex functions, exponential-convex functions, exponential-type convex functions, refined exponential-type convex functions, n-polynomial convex functions, $\sigma ,s$-convex functions, modified $(p,h)$-convex functions, co-ordinated-convex functions, relative-convex functions, quasi-convex functions, $(\alpha ,h-m)-p$-convex functions, and preinvex functions. Included in the fractional integral operators are Riemann–Liouville (R-L) fractional integral, Katugampola fractional integral, k-R-L fractional integral, $(k,s)$-R-L fractional integral, Caputo-Fabrizio (C-F) fractional integral, R-L fractional integrals of a function with respect to another function, Hadamard fractional integral, and Raina fractional integral operator. Full article

In this paper, we provide different variants of the Hermite–Hadamard ($H\cdot H$) inequality using the concept of a new class of convex mappings, which is referred to as up and down harmonically $s$-convex fuzzy-number-valued functions ($U\mathcal{D}\mathcal{H}$ $s$-convex $\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$) in the second sense based on the up and down fuzzy inclusion relation. The findings are confirmed with certain numerical calculations that take a few appropriate examples into account. The results deal with various integrals of the $\frac{2\rho \sigma }{\rho +\sigma }$ type and are innovative in the setting of up and down harmonically $s$-convex fuzzy-number-valued functions. Moreover, we acquire classical and new exceptional cases that can be seen as applications of our main outcomes. In our opinion, this will make a significant contribution to encouraging more research. Full article

Many authors have obtained some inclusion properties of certain subclasses of univalent and functions associated with distribution series, such as Pascal distribution, Binomial distribution, Poisson distribution, Mittag–Leffler-type Poisson distribution, and Geometric distribution. In the present paper, we obtain some inclusion relations of the harmonic class $\mathcal{H}(\alpha ,\delta )$ with the classes ${\mathcal{S}}_{\mathcal{H}}^{*}$ of starlike harmonic functions and ${\mathcal{K}}_{\mathcal{H}}$ of convex harmonic functions, also for the harmonic classes ${\mathcal{TN}}_{\mathcal{H}}\left(\beta \right)$ and ${\mathcal{TR}}_{\mathcal{H}}\left(\beta \right)$ associated with the operator $\mathsf{{\rm Y}}$ defined by applying certain convolution operator regarding Poisson distribution series. Several consequences and corollaries of the main results are also obtained. Full article

Integral inequalities have accumulated a comprehensive and prolific field of research within mathematical interpretations. In recent times, strategies of fractional calculus have become the subject of intensive research in historical and contemporary generations because of their applications in various branches of science. In this paper, we concentrate on establishing Hermite–Hadamard and Pachpatte-type integral inequalities with the aid of two different fractional operators. In particular, we acknowledge the critical Hermite–Hadamard and related inequalities for n-polynomial s-type convex functions and n-polynomial s-type harmonically convex functions. We practice these inequalities to consider the Caputo–Fabrizio and the k-Riemann–Liouville fractional integrals. Several special cases of our main results are also presented in the form of corollaries and remarks. Our study offers a better perception of integral inequalities involving fractional operators. Full article

The main objective of this paper is to establish some new variants of the Jensen–Mercer inequality via harmonically strongly convex function. We also propose some new fractional analogues of Hermite–Hadamard–Jensen–Mercer-like inequalities using $\mathcal{AB}$ fractional integrals. In order to obtain some of our main results, we also derive new fractional integral identities. To demonstrate the significance of our main results, we present some interesting applications to special means and to error bounds as well. Full article