Search Results (30)

In this paper, a novel class of meromorphic functions associated with the Mittag–Leffler function $E_{\mu ,\vartheta }\left(z\right)$ is introduced using the Hilbert space operator. In the punctured symmetric domain ${⋓}^{\ast }$, essential properties of this class are systematically investigated. These properties include coefficient inequalities, growth and distortion bounds, as well as weighted and arithmetic mean estimates. Furthermore, the extreme points and radii of geometric properties such as close-to-convexity, starlikeness, and convexity are analyzed in detail. Additionally, the Hadamard product (or convolution) is explored to demonstrate the algebraic structure and stability of the introduced function class under this operation. Integral mean inequalities are also established to provide further insights into the behavior of these functions within the given domain. Full article

We introduce and systematically study a new class $k^{\lambda ,p}(\alpha ,\beta )$ of meromorphic p-valent functions defined by means of the Ruscheweyh-type operator $D_{*}^{\lambda ,p}$, where $p\in \mathbb{N}$, $\lambda >-p$, $0\le \alpha <1$, and $\beta >0$. Membership in this class is characterized through coefficient estimates. Also investigated are growth, distortion, stability under convex combinations, radii of starlikeness and convexity of order $\rho$$(0\le \rho <1)$, convolution, the action of an integral operator of Bernardi–Libera–Livingston type, and neighborhoods. Full article

In this effort, we extend the fractal–fractional operators into the complex plane together with the quantum calculus derivative to obtain a quantum–fractal–fractional operators (QFFOs). Using this newly created operator, we create an entirely novel subclass of analytical functions in the unit disk. Motivated by the concept of differential subordination, we explore the most important geometric properties of this novel operator. This leads to a study on a set of differential inequalities in the open unit disk. We focus on the conditions to obtain a bounded turning function of QFFOs. Some examples are considered, involving special functions like Bessel and generalized hypergeometric functions. Full article

Restricted by a metal-enclosed structure, the internal defects of large transformers are difficult to visually detect. In this paper, a micro-robot is used to visually inspect the interior of a transformer. For the micro-robot to successfully detect the discharge level and insulation degradation trend in the transformer, it is essential to segment the carbon trace accurately and rapidly from the complex background. However, the complex edge features and significant size differences of carbon traces pose a serious challenge for accurate segmentation. To this end, we propose the Hadamard production-Spatial coordinate attention-PixelShuffle UNet (HSP-UNet), an innovative architecture specifically designed for carbon trace segmentation. To address the pixel over-concentration and weak contrast of carbon trace image, the Adaptive Histogram Equalization (AHE) algorithm is used for image enhancement. To realize the effective fusion of carbon trace features with different scales and reduce model complexity, the novel grouped Hadamard Product Attention (HPA) module is designed to replace the original convolution module of the UNet. Meanwhile, to improve the activation intensity and segmentation completeness of carbon traces, the Spatial Coordinate Attention (SCA) mechanism is designed to replace the original jump connection. Furthermore, the PixelShuffle up-sampling module is used to improve the parsing ability of complex boundaries. Compared with UNet, UNet++, UNeXt, MALUNet, and EGE-UNet, HSP-UNet outperformed all the state-of-the-art methods on both carbon trace datasets. For dendritic carbon traces, HSP-UNet improved the Mean Intersection over Union (MIoU), Pixel Accuracy (PA), and Class Pixel Accuracy (CPA) of the benchmark UNet by 2.13, 1.24, and 4.68 percentage points, respectively. For clustered carbon traces, HSP-UNet improved MIoU, PA, and CPA by 0.98, 0.65, and 0.83 percentage points, respectively. At the same time, the validation results showed that the HSP-UNet has a good model lightweighting advantage, with the number of parameters and GFLOPs of 0.061 M and 0.066, respectively. This study could contribute to the accurate segmentation of discharge carbon traces and the assessment of the insulation condition of the oil-immersed transformer. Full article

Knowledge graph embedding (KGE) has been identified as an effective method for link prediction, which involves predicting missing relations or entities based on existing entities or relations. KGE is an important method for implementing knowledge representation and, as such, has been widely used in driving intelligent applications w.r.t. question-answering systems, recommendation systems, and relationship extraction. Models based on convolutional neural networks (CNNs) have achieved good results in link prediction. However, as the coverage areas of knowledge graphs expand, the increasing volume of information significantly limits the performance of these models. This article introduces a triple-attention-based multi-channel CNN model, named ConvAMC, for the KGE task. In the embedding representation module, entities and relations are embedded into a complex space and the embeddings are performed in an alternating pattern. This approach helps in capturing richer semantic information and enhances the expressive power of the model. In the encoding module, a multi-channel approach is employed to extract more comprehensive interaction features. A triple attention mechanism and max pooling layers are used to ensure that interactions between spatial dimensions and output tensors are captured during the subsequent tensor concatenation and reshaping process, which allows preserving local and detailed information. Finally, feature vectors are transformed into prediction targets for embedding through the Hadamard product of feature mapping and reshaping matrices. Extensive experiments were conducted to evaluate the performance of ConvAMC on three benchmark datasets compared with state-of-the-art (SOTA) models, demonstrating that the proposed model outperforms all compared models across all evaluation metrics on two of the datasets, and achieves advanced link prediction results on most evaluation metrics on the third dataset. Full article

In the current paper, we introduce new subclasses of analytic and bi-univalent functions involving Caputo-type fractional derivatives subordinating to the Lucas polynomial. Furthermore, we find non-sharp estimates on the first two Taylor–Maclaurin coefficients $\left|a_2\right|$ and $\left|a_3\right|$ for functions in these subclasses. Using the values of $\left|a_2\right|$ and $\left|a_3\right|$, we determined Fekete–Szegő inequality for functions in these subclasses. Moreover, we pointed out some more subclasses by fixing the parameters involved in Lucas polynomial and stated the estimates and Fekete–Szegő inequality results without proof. Full article

Certain characteristics of univalent functions with negative coefficients of the form $f\left(z\right)=z-{\sum }_{n=1}^{\infty }{a}_{2n}{z}^{2n},a_{2n}>0$ have been studied by Silverman and Berman. Pokley, Patil and Shrigan have discovered some insights into the Hadamard product of $\mathcal{P}$-valent functions with negative coefficients. S. M. Khairnar and Meena More have obtained coefficient limits and convolution results for univalent functions lacking a alternating type coefficient. In this paper, using the $\mathfrak{q}$-Difference operator, we developed the a subclass of meromorphically $\mathcal{P}$-valent functions with alternating coefficients. Additionally, we obtained multivalent function convolution results and coefficient limits. Full article

In this article, we first define and then propose to systematically study some new subclasses of the class of analytic and bi-concave functions in the open unit disk. For this purpose, we make use of a combination of the binomial series and the confluent hypergeometric function. Among some other properties and results, we derive the estimates on the initial Taylor-Maclaurin coefficients $|a_2|$ and $|a_3|$ for functions in these analytic and bi-concave function classes, which are introduced in this paper. We also derive a number of corollaries and consequences of our main results in this paper. Full article

Using the Salagean q-differential operator, we investigate a novel subclass of analytic functions in the open unit disc, and we use the Hadamard product to provide some inclusion relations. Furthermore, the coefficient conditions, convolution properties, and applications of the q-fractional calculus operators are investigated for this class of functions. In addition, we extend the Miller and Mocanu inequality to the q-theory of analytic functions. Full article

In this article, a new linear extended multiplier operator is defined utilizing the q-Choi–Saigo–Srivastava operator and the q-derivative. Two generalized subclasses of q—uniformly convex and starlike functions of order $\delta$—are defined and studied using this new operator. Necessary conditions are derived for functions to belong in each of the two subclasses, and subordination theorems involving the Hadamard product of such particular functions are stated and proven. As applications of those findings using specific values for the parameters of the new subclasses, associated corollaries are provided. Additionally, examples are created to demonstrate the conclusions’ applicability in relation to the functions from the newly introduced subclasses. Full article

The potential for widespread applications of the geometric and mapping properties of functions of a complex variable has motivated this article. On the other hand, the basic or quantum (or q-) derivatives and the basic or quantum (or q-) integrals are extensively applied in many different areas of the mathematical, physical and engineering sciences. Here, in this article, we first apply the q-calculus in order to introduce the q-derivative operator ${\mathfrak{S}}_{\eta ,p,q}^{n,m}$. Secondly, by means of this q-derivative operator, we define an interesting subclass $\mathcal{T}{\aleph }_{\lambda ,p}^{n,m}(\eta ,\alpha ,\kappa )$ of the class of normalized analytic and multivalent (or p-valent) functions in the open unit disk $\mathbb{U}$. This p-valent analytic function class is associated with the class $\kappa$-$\mathcal{UCV}$ of $\kappa$-uniformly convex functions and the class $\kappa$-$\mathcal{UST}$ of $\kappa$-uniformly starlike functions in $\mathbb{U}$. For functions belonging to the normalized analytic and multivalent (or p-valent) function class $\mathcal{T}{\aleph }_{\lambda ,p}^{n,m}(\eta ,\alpha ,\kappa )$, we then investigate such properties as those involving (for example) the coefficient bounds, distortion results, convex linear combinations, and the radii of starlikeness, convexity and close-to-convexity. We also consider a number of corollaries and consequences of the main findings, which we derived herein. Full article

In this article, we make use of the q-binomial theorem to introduce and study two new subclasses $\aleph (\alpha q,q)$ and $\aleph (\alpha ,q)$ of meromorphic functions in the open unit disk $\mathbb{U}$, that is, analytic functions in the punctured unit disk ${\mathbb{U}}^{\ast }=\mathbb{U}\backslash \left\{0\right\}=\{z:z\in \mathbb{C}\mathrm{and}0<\left|z\right|<1\}.$ We derive inclusion relations and investigate an integral operator that preserves functions which belong to these function classes. In addition, we establish a strict inequality involving a certain linear convolution operator which we introduce in this article. Several special cases and corollaries of our main results are also considered. Full article

In this article, we introduce and study the behavior of the modules of the first two coefficients for the classes ${\mathcal{N}}_{\mathsf{\Sigma }}(\gamma ,\lambda ,\delta ,\mu ;\alpha )$ and ${\mathcal{N}}_{{\mathsf{\Sigma }}^{*}}(\gamma ,\lambda ,\delta ,\mu ;\beta )$ of normalized holomorphic and bi-univalent functions that are connected with the prestarlike functions. We determine the upper bounds for the initial Taylor–Maclaurin coefficients $|a_2|$ and $|a_3|$ for the functions of each of these families, and we also point out some special cases and consequences of our main results. The study of these classes is closely connected with those of Ruscheweyh who in 1977 introduced the classes of prestarlike functions of order $\mu$ using a convolution operator and the proofs of our results are based on the well-known Carathédory’s inequality for the functions with real positive part in the open unit disk. Our results generalize a few of the earlier ones obtained by Li and Wang, Murugusundaramoorthy et al., Brannan and Taha, and could be useful for those that work with the geometric function theory of one-variable functions. Full article

In this research, a novel linear operator involving the Borel distribution and Mittag-Leffler functions is introduced using Hadamard products or convolutions. This operator is utilized to develop new subfamilies of bi-univalent functions via the principle of subordination with Gegenbauer orthogonal polynomials. The investigation also focuses on the estimation of the coefficients $|a_{\ell }|(\ell =2,3)$ and the Fekete–Szegö inequality for functions belonging to these subfamilies of bi-univalent functions. Several corollaries and implications of the findings are discussed. Overall, this study presents a new approach for constructing bi-univalent functions and provides valuable insights for further research in this area. Full article

For the first time, we attempted to define two new sub-classes of bi-univalent functions in the open unit disc of the complex order involving Mathieu-type series, associated with generalized telephone numbers. The initial coefficients of functions in these classes were obtained. Moreover, we also determined the Fekete–Szegö inequalities for function in these and several related corollaries. Full article