Search Results (29)

In this paper, necessary and sufficient efficiency conditions in new multi-cost variational models are formulated and proved. To this end, we introduce a new notion of $({\vartheta }_0,{\vartheta }_1)-({\sigma }_0,{\sigma }_1)-type-I$ functionals determined by multiple integrals. To better emphasize the significance of the suggested $({\vartheta }_0,{\vartheta }_1)-({\sigma }_0,{\sigma }_1)-type-I$ functionals and how they add to previous studies, we mention that the $({\vartheta }_0,{\vartheta }_1)-({\sigma }_0,{\sigma }_1)-type-I$ and generalized $({\vartheta }_0,{\vartheta }_1)-({\sigma }_0,{\sigma }_1)-type-I-typeI$ assumptions associated with the involved multiple integral functionals cover broader and more general classes of problems, where the convexity of the functionals is not fulfilled or the functionals considered are not of simple integral type. In addition, innovative proofs are provided for the main results. Full article

This paper investigates the analytical properties of multiplicative generalized proportional $\sigma$-Riemann–Liouville fractional integrals and the corresponding Hermite–Hadamard-type inequalities. Central to our study are two key notions: multiplicative $\sigma$-convex functions and multiplicative generalized proportional $\sigma$-Riemann–Liouville fractional integrals, both of which serve as the foundational framework for our analysis. We first introduce and examine several fundamental properties of the newly defined fractional integral operator, including continuity, commutativity, semigroup behavior, and boundedness. Building on these results, we derive a novel identity involving this operator, which forms the basis for establishing new Hermite–Hadamard-type inequalities within the multiplicative setting. To validate the theoretical results, we provide multiple illustrative examples and perform graphical visualizations. These examples not only demonstrate the correctness of the derived inequalities but also highlight the practical relevance and potential applications of the proposed framework. Full article

In this article, the strong class of bi-close-to-convex functions of order $\alpha$ and $\beta$ in n-fold symmetric bi-univalent functions, which is the subclass of $\sigma$, is introduced. The upper bound value for $\left|a_{n+1}\right|$, $\left|a_{2n+1}\right|$ for functions in these classes are obtained. Moreover, the Fekete–Szegö relation for our classes of functions are established. Full article

In this research, we demonstrate novel Hermite–Hadamard–Mercer fractional integral inequalities using a wide class of fractional integral operators (the Raina fractional operator). Moreover, a new lemma of this type is proved, and new identities are obtained using the definition of convex function. In addition to a detailed derivation of a few special situations, certain known findings are summarized. We also point out that some results in this study, in some special cases, such as setting $\alpha =0=\phi ,\gamma =1,$ and $w=0,\sigma \left(0\right)=1,\lambda =1$, are more reasonable than those obtained. Finally, it is believed that the technique presented in this paper will encourage additional study in this field. Full article

The geometric properties of q-Bessel and q-Bessel-Struve functions are examined in this study. For each of them, three different normalizations are applied in such a way that the resulting functions are analytic in the unit disk of the complex plane. For these normalized functions, the radii of $\gamma$-spirallike and convex $\gamma$-spirallike of order $\sigma$ are determined using their Hadamard factorization. These findings extend the known results for Bessel and Struve functions. The characterization of entire functions from the Laguerre-Pólya class plays an important role in our proofs. Additionally, the interlacing property of zeros of q-Bessel and q-Bessel-Struve functions and their derivatives is useful in the proof of our main theorems. Full article

In this article, we first consider the fractional q-differential operator and the $\left(\lambda ,q\right)$-fractional differintegral operator $D_q^{\lambda }:\mathcal{A}\to \mathcal{A}$. Using the $\left(\lambda ,q\right)$-fractional differintegral operator, we define two new subclasses of analytic functions: the subclass ${\mathcal{S}}^{*}\left(q,\beta ,\lambda \right)$ of starlike functions of order $\beta$ and the class $C_{\mathrm{\Sigma }}^{\lambda ,q}\left(\alpha \right)$ of bi-close-to-convex functions of order $\beta$. We explore the results on coefficient inequality and Fekete–Szegö problems for functions belonging to the class ${\mathcal{S}}^{*}\left(q,\beta ,\lambda \right)$. Using the Faber polynomial technique, we derive upper bounds for the nth coefficient of functions in the class of bi-close-to-convex functions of order $\beta$. We also investigate the erratic behavior of the initial coefficients in the class $C_{\mathrm{\Sigma }}^{\lambda ,q}\left(\alpha \right)$ of bi-close-to-convex functions. Furthermore, we address some known problems to demonstrate the connection between our new work and existing research. Full article

In the present paper, utilizing a wide class of fractional integral operators (namely the Raina fractional operator), we develop novel fractional integral inequalities of the Hermite–Hadamard type. With the help of the well-known Riemann–Liouville fractional operators, s-type convex functions are derived using the important results. We also note that some of the conclusions of this study are more reasonable than those found under certain specific conditions, e.g., $s=1$, $\lambda =\alpha ,$ $\sigma \left(0\right)=1,$ and $w=0$. In conclusion, the methodology described in this article is expected to stimulate further research in this area. Full article

We consider the large class $\mathfrak{G}$ of locally convex spaces that includes, among others, the classes of $\left(DF\right)$-spaces and $\left(LF\right)$-spaces. For a space E in class $\mathfrak{G}$ we have characterized that a subspace Y of $(E,\sigma (E,E^{\prime }))$, endowed with the induced topology, is analytic if and only if Y has a $\sigma (E,E^{\prime })$-compact resolution and is contained in a $\sigma (E,E^{\prime })$-separable subset of E. This result is applied to reprove a known important result (due to Cascales and Orihuela) about weak metrizability of weakly compact sets in spaces of class $\mathfrak{G}$. The mentioned characterization follows from the following analogous result: The space $C\left(X\right)$ of continuous real-valued functions on a completely regular Hausdorff space X endowed with a topology $\xi$ stronger or equal than the pointwise topology ${\tau }_p$ of $C\left(X\right)$ is analytic iff $\left(C\right(X),\xi )$ is separable and is covered by a compact resolution. Full article

In current manuscript, using Laguerre polynomials and $(p-q)$-Wanas operator, we identify upper bounds $\left|a_2\right|$ and $\left|a_3\right|$ which are first two Taylor-Maclaurin coefficients for a specific bi-univalent functions classes ${\mathcal{W}}_{\sum }(\eta ,\delta ,\lambda ,\sigma ,\theta ,\alpha ,\beta ,p,q;h)$ and ${\mathcal{K}}_{\sum }(\xi ,\rho ,\sigma ,\theta ,\alpha ,\beta ,p,q;h)$ which cover the convex and starlike functions. Also, we discuss Fekete-Szegö type inequality for defined class. 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

This research aims to present a linear operator ${\mathcal{L}}_{p,q}^{\rho ,\sigma ,\mu }f$ utilizing the q-Mittag–Leffler function. Then, we introduce the subclass of harmonic $(p,q)$-convex functions ${\mathcal{HT}}_{p,q}(\vartheta ,\mathcal{W},\mathcal{V})$ related to the Janowski function. For the harmonic p-valent functions f class, we investigate the harmonic geometric properties, such as coefficient estimates, convex linear combination, extreme points, and Hadamard product. Finally, the closure property is derived using the subclass ${\mathcal{HT}}_{p,q}(\vartheta ,\mathcal{W},\mathcal{V})$ under the q-Bernardi integral operator. Full article

In this study, we introduce new generalizations of higher-order type-I functions and higher-order pseudo-convexity type-I functions. The application of the notion of sublinear functionals to these generalizations of higher-order type-I and higher-order pseudo-convexity type-I functions is crucial to our main findings. Furthermore, under these generalizations of the higher-order type-I and higher-order pseudo-convexity type-I functions, we established and studied six new types of higher-order duality models and programs for multiple objective nonlinear programming problems. In addition, we use these generalizations of higher-order type-I functions and higher-order pseudo-convexity type-I functions, to formulate and prove the theorems of weak duality, strong duality, and strict converse duality for these new six types of higher-order model programs. Full article

In the present paper, due to beta negative binomial distribution series and Laguerre polynomials, we investigate a new family ${\mathcal{F}}_{\mathsf{\Sigma }}(\delta ,\eta ,\lambda ,\theta ;h)$ of normalized holomorphic and bi-univalent functions associated with Ozaki close-to-convex functions. We provide estimates on the initial Taylor–Maclaurin coefficients and discuss Fekete–Szegő type inequality for functions in this family. Full article

In this paper, we extend some Steffensen-type inequalities to time scales by using the diamond-$\alpha$-dynamic integral. Further, we prove some new Steffensen-type inequalities for convex functions utilizing positive $\sigma$-finite measures in time scale calculus. Moreover, as a special case, we obtain these inequalities for the delta and the nabla integral. By using the relation between calculus on time scales $\mathbb{T}$ and differential calculus on $\mathbb{R}$, we obtain already-known Steffensen-type inequalities. Full article