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In this paper, we define a new class $ \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right) $ of holomorphic functions in the open unit disk defined connected with the combination binomial series and Babalola operator using the differential subordination with Janowski-type functions. Using the well-known Carathéodory's inequality for function with real positive parts and the Keogh-Merkes and Ma-Minda's in equalities, we determined the upper bound for the first two initial coefficients of the Taylor-Maclaurin power series expansion. Then, we found an upper bound for the Fekete-Szegö functional for the functions in this family. Further, a similar result for the first two coefficients and for the Fekete-Szegő inequality have been done the function $ \mathcal{G}^{-1} $ when $ \mathcal{G}\in \mathcal{R} _{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right) $. Next, for the functions of these newly defined family we determine coefficient estimates, distortion bounds, radius problems, and the radius of starlikeness and close-to-convexity. The novelty of the results is that we were able to investigate basic properties of these new classes of functions using simple methods and these classes are connected with the new convolution operator and the Janowski functions.

In this article, using the Sǎlǎgean operator, we introduced three new subclasses of bi-univalent functions associated with bounded boundary rotation in open unit disk $ \mathbb{E}. $ For these new classes, we first obtain initial Taylor-Maclaurin's coefficient bounds. Furthermore, the famous Fekete-Szegö inequality was also derived for these new subclass functions. Some improved results, when compared with those available in the literature, are also stated.

Our aim was to develop a new class of bi starlike functions by utilizing the concept of subordination, driven by the idea of multiplicative calculus, specifically multiplicative derivatives. Several restrictions were imposed, which were indeed strict constraints, because we have tried to work within the current framework or the design of analytic functions. To make the study more versatile, we redefined our new class of function with Miller-Ross Poisson distribution (MRPD), in order to increase the study's adaptability. We derived the first coefficient estimates and Fekete-Szegő inequalities for functions in this new class. To demonstrate the characteristics, we have provided a few examples.

In the present paper, we introduce a new subclass of normalized analytic and univalent functions in the open unit disk associated with Sigmoid function. Coefficient estimates, convolution conditions, convexity and some other geometric properties for functions in this class are investigated. Also, subordination and inclusion results are obtained.

In recent years, the concept of fuzzy set has been incorporated into the field of geometric function theory, leading to the evolution of the classical concept of differential subordination into that of fuzzy differential subordination. In this study, certain generalized classes of $ p $ -valent analytic functions are defined in the context of fuzzy subordination. It is highlighted that for particular functions used in the definitions of those classes, the classes of fuzzy $ p $-valent convex and starlike functions are obtained, respectively. The new classes are introduced by using a $ \mathfrak{q} $-calculus operator defined in this investigation using the concept of convolution. Some inclusion results are discussed concerning the newly introduced classes based on the means given by the fuzzy differential subordination theory. Furthermore, connections are shown between the important results of this investigation and earlier ones. The second part of the investigation concerns a new generalized $ \mathfrak{q} $-calculus operator, defined here and having the $ (p, \mathfrak{q)} $-Bernardi operator as particular case, applied to the functions belonging to the new classes introduced in this study. Connections between the classes are established through this operator.

Abstract Recently, El-Deeb and Cotîrlă (Mathematics 11:11234834, 2023) used the error function together with a q-convolution to introduce a new operator. By means of this operator the following class R α , ϒ λ , q ( δ , η ) $\mathcal{R}_{\alpha ,\Upsilon}^{\lambda ,q}(\delta ,\eta )$ of analytic functions was studied: R α , ϒ λ , q ( δ , η ) : = { F : ℜ ( ( 1 − δ + 2 η ) H ϒ λ , q F ( ζ ) ζ + ( δ − 2 η ) ( H ϒ λ , q F ( ζ ) ) ′ + η ζ ( H ϒ λ , q F ( ζ ) ) ″ ) } > α ( 0 ≦ α < 1 ) . $$\begin{aligned} &\mathcal{R}_{\alpha ,\Upsilon }^{\lambda ,q}(\delta ,\eta ) \\ &\quad := \biggl\{ \mathcal{ F}: {\Re} \biggl( (1-\delta +2\eta ) \frac{\mathcal{H}_{\Upsilon }^{\lambda ,q}\mathcal{F}(\zeta )}{\zeta}+(\delta -2\eta ) \bigl(\mathcal{H} _{\Upsilon}^{\lambda ,q}\mathcal{F}(\zeta ) \bigr) ^{{ \prime}}+\eta \zeta \bigl( \mathcal{H}_{\Upsilon}^{\lambda ,q} \mathcal{F}( \zeta ) \bigr) ^{{{\prime \prime}}} \biggr) \biggr\} \\ &\quad >\alpha \quad (0\leqq \alpha < 1). \end{aligned}$$ For these general analytic functions F ∈ R β , ϒ λ , q ( δ , η ) $\mathcal{F}\in \mathcal{R}_{\beta ,\Upsilon}^{\lambda ,q}(\delta , \eta )$ , we give upper bounds for the Fekete–Szegö functional and for the second and third Hankel determinants.

These days, integral equations govern a wide range of physical processes, including those related to electricity, mechanics, fluid dynamics, ecology, soil moisture dynamics, shallow water wave propagation, and chemical science. As a result, creating new techniques and putting them into practice for solving integral equations becomes more crucial. In this paper, the Upadhyaya integral transform is employed for determining the analytical solutions of the non-linear Volterra integral equations (NVIEs) of the first kind. Four examples suggest the effectiveness of the Upadhyaya integral transform, particularly for NVIEs of the first kind. The calculation results suggest that the proposed method provides accurate solutions to the original problem and this method is a valuable tool for researchers and scientists working on the broader range of problems involving NVIEs of the first kind.

The research here was motivated by a number of recent studies on Hankel inequalities and sharp bounds. In this article, we define a new subclass of holomorphic convex functions that are related to tangent functions. We then derive geometric properties like the necessary and sufficient conditions, radius of convexity, growth, and distortion estimates for our defined function class. Furthermore, the sharp coefficient bounds, sharp Fekete-Szegö inequality, sharp 2nd order Hankel determinant, and Krushkal inequalities are given. Moreover, we calculate the sharp coefficient bounds, sharp Fekete-Szegö inequality, and sharp second-order Hankel determinant for the functions whose coefficients are logarithmic.

Recently, El-Deeb and Cotîrlă (Mathematics 11:11234834, 2023) used the error function together with a q-convolution to introduce a new operator. By means of this operator the following class R α , ϒ λ , q (δ , η) of analytic functions was studied: R α , ϒ λ , q (δ , η) : = { F : ℜ ((1 − δ + 2 η) H ϒ λ , q F (ζ) ζ + (δ − 2 η) (H ϒ λ , q F (ζ)) ′ + η ζ (H ϒ λ , q F (ζ)) ″) } > α (0 ≦ α < 1). For these general analytic functions F ∈ R β , ϒ λ , q (δ , η) , we give upper bounds for the Fekete–Szegö functional and for the second and third Hankel determinants. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce a new subclass of normal- ized analytic and univalent functions in the open unit disk associated with Sigmoid function. Coeficient estimates, convolution conditions, convex- ity and some other geometric properties for functions in this class are investigated. Also, subordination and inclusion results are obtained. [ABSTRACT FROM AUTHOR]

In this study, we define new class of holomorphic functions associated with tangent function. Furthermore, we examine the differential subordination implementation results related to Janowski and tangent functions. Also, we investigate some extreme point theorem and partial sums results, necessary and sufficient conditions, convex combination, closure theorem, growth and distortion bounds, and radii of close-to-starlikeness and starlikeness for this newly defined functions class of holomorphic functions.

Abstract The present investigation aims to examine the geometric properties of the normalized form of the combination of generalized Lommel–Wright function J λ , μ ν , m ( z ) : = Γ m ( λ + 1 ) Γ ( λ + μ + 1 ) 2 2 λ + μ z 1 − ( ν / 2 ) − λ J λ , μ ν , m ( z ) $\mathfrak{J}_{\lambda ,\mu}^{\nu ,m}(z):=\Gamma ^{m}(\lambda +1) \Gamma (\lambda +\mu +1)2^{2\lambda +\mu}z^{1-(\nu /2)-\lambda} \mathcal{J}_{\lambda ,\mu }^{\nu ,m}(\sqrt{z})$ , where the function J λ , μ ν , m $\mathcal{J}_{\lambda ,\mu}^{\nu ,m}$ satisfies the differential equation J λ , μ ν , m ( z ) : = ( 1 − 2 λ − ν ) J λ , μ ν , m ( z ) + z ( J λ , μ ν , m ( z ) ) ′ $\mathcal{J}_{\lambda ,\mu}^{\nu ,m}(z):=(1-2\lambda -\nu )J_{ \lambda ,\mu}^{\nu ,m}(z)+z (J_{\lambda ,\mu }^{\nu ,m}(z) )^{\prime}$ with J ν , λ μ , m ( z ) = ( z 2 ) 2 λ + ν ∑ k = 0 ∞ ( − 1 ) k Γ m ( k + λ + 1 ) Γ ( k μ + ν + λ + 1 ) ( z 2 ) 2 k $$ J_{\nu ,\lambda}^{\mu ,m}(z)= \biggl(\frac{z}{2} \biggr)^{2\lambda + \nu} \sum_{k=0}^{\infty} \frac{(-1)^{k}}{\Gamma ^{m} (k+\lambda +1 )\Gamma (k\mu +\nu +\lambda +1 )} \biggl(\frac{z}{ 2} \biggr)^{2k} $$ for λ ∈ C ∖ Z − $\lambda \in \mathbb{C}\setminus \mathbb{Z}^{-}$ , Z − : = { − 1 , − 2 , − 3 , … } $\mathbb{Z}^{-}:= \{ -1,-2,-3,\ldots \}$ , m ∈ N $m\in \mathbb{N}$ , ν ∈ C $\nu \in \mathbb{C}$ , and μ ∈ N 0 : = N ∪ { 0 } $\mu \in \mathbb{N}_{0}:=\mathbb{N}\cup \{0\}$ . In particular, we employ a new procedure using mathematical induction, as well as an estimate for the upper and lower bounds for the gamma function inspired by Li and Chen (J. Inequal. Pure Appl. Math. 8(1):28, 2007), to evaluate the starlikeness and convexity of order α, 0 ≤ α < 1 $0\leq \alpha

The present investigation aims to examine the geometric properties of the normalized form of the combination of generalized Lommel–Wright function J λ , μ ν , m (z) : = Γ m (λ + 1) Γ (λ + μ + 1) 2 2 λ + μ z 1 − (ν / 2) − λ J λ , μ ν , m (z) , where the function J λ , μ ν , m satisfies the differential equation J λ , μ ν , m (z) : = (1 − 2 λ − ν) J λ , μ ν , m (z) + z (J λ , μ ν , m (z)) ′ with J ν , λ μ , m (z) = (z 2) 2 λ + ν ∑ k = 0 ∞ (− 1) k Γ m (k + λ + 1) Γ (k μ + ν + λ + 1) (z 2) 2 k for λ ∈ C ∖ Z − , Z − : = { − 1 , − 2 , − 3 , ... } , m ∈ N , ν ∈ C , and μ ∈ N 0 : = N ∪ { 0 } . In particular, we employ a new procedure using mathematical induction, as well as an estimate for the upper and lower bounds for the gamma function inspired by Li and Chen (J. Inequal. Pure Appl. Math. 8(1):28, 2007), to evaluate the starlikeness and convexity of order α, 0 ≤ α < 1 . Ultimately, we discuss the starlikeness and convexity of order zero for J λ , μ ν , m , and it turns out that they are useful to extend the range of validity for the parameter λ to λ ≥ 0 where the main concept of the proofs comes from some technical manipulations given by Mocanu (Libertas Math. 13:27–40, 1993). Our results improve, complement, and generalize some well-known (nonsharp) estimates. [ABSTRACT FROM AUTHOR]

In this paper, we examine a type of constant elasticity of variance model that is subject to its terminal condition. We prove that certain transformations may be applied to obtain a simpler equation that has known solution processes. Four cases are obtained that play a role in specifying the many unknown parameters of the model. The corresponding terminal condition is transformed into an initial condition, and we then demonstrate how to solve this Cauchy problem by using Lie symmetries and Poisson's formula. Finally, we examine the behaviour of the obtained solutions.

The Geometric Theory of Analytic Functions was initially developed for the space of functions that are analytic in the unit disk. The convexity and starlikeness of functions are the first geometric ideas considered in this theory. We can notice a symmetry between the subjects considered in the space of analytic functions and those in the space of harmonic functions. In the presented paper, we consider the starlikeness of functions in the space of meromorphic harmonic functions.

In this paper, we introduce new classes of functions defined within the open unit disk by integrating the concepts of (λ,γ)-symmetrical functions, generalized Janowski functions, and quantum calculus. We derive a structural formula and a representation theorem for the class Sqλ,γ(x,y,z). Utilizing convolution techniques and quantum calculus, we investigate convolution conditions supported by examples and corollary, establishing sufficient conditions. Additionally, we derive properties related to coefficient estimates, which further elucidate the characteristics of the defined function classes.

By using the Mittag–Leffler function associated with functions of bounded boundary rotation, the authors introduce a few new subclasses of bi-univalent functions involving the Mittag–Leffler function with bounded boundary rotation in the open unit disk D. For these new classes, the authors establish initial coefficient bounds of |a2| and |a3|. Furthermore, the famous Fekete–Szegö coefficient inequality is also obtained for these new classes of functions.

This paper establishes new results related to geometric function theory by presenting a new subclass of harmonic functions with complex values within the open unit disk, characterized by a second-order differential inequality. The investigation explores the bounds on the coefficients and estimates of the function growth. This paper also demonstrates that this subclass remains stable under the convolution operation applied to its members. In addition, in the last section, images of the unit disk under some functions of this class are given.

A challenging part of studying geometric function theory is figuring out the sharp boundaries for coefficient-related problems that crop up in the Taylor–Maclaurin series of univalent functions. Using Caputo-type fractional derivatives to define the families of Sakaguchi-type starlike functions with respect to symmetric points, this article aims to investigate the first three initial coefficient estimates, the bounds for various problems such as Fekete–Szegő inequality, and the Zalcman inequalities, by subordinating to the function of the three leaves domain. Fekete–Szegő-type inequalities and initial coefficients for functions of the form H−1 and ζH(ζ) and 12logHζζ connected to the three leaves functions are also discussed.

In the present investigation, we introduce a new subclass of univalent functions F(u,λ) and a subclass of bi-univalent function Fo,Σ(u,λ) with bounded boundary and bounded radius rotation. Some examples of the functions belonging to the classes F(u,λ) are also derived. For these new classes, the authors derive many interesting relations between these classes and the existing familiar subclasses in the literature. Furthermore, the authors establish new coefficient estimates for these classes. Apart from the above, the first two initial coefficient bounds for the class Fo,Σ(u,λ) are established.

The topic concerning the introduction and investigation of new classes of analytic functions using subordination techniques for obtaining certain geometric properties alongside coefficient estimates and inclusion relations is enriched by the results of the present investigation. The prolific tools of quantum calculus applied in geometric function theory are employed for the investigation of a new class of analytic functions introduced by applying a previously defined generalized q−integral operator and the concept of subordination. Investigations are conducted on the new class, including coefficient estimates, integral representation for the functions of the class, linear combinations, forms of the weighted and arithmetic means involving functions from the class, and the estimation of the integral means results.

Convolution plays a pivotal role in the domains of signal processing and optics. This paper primarily focuses on studying the weighted convolution for quaternion linear canonical cosine transform (QLCcT) and its application in multiplicative filter analysis. Firstly, we propose QLCcT by combining quaternion algebra with linear canonical cosine transform (LCcT), which extends LCcT to Hamiltonian quaternion algebra. Secondly, we introduce weighted convolution and correlation operations for QLCcT, accompanied by their corresponding theorems. We also explore the properties of QLCcT. Thirdly, we utilize these proposed convolution structures to analyze multiplicative filter models that offer lower computational complexity compared to existing methods based on quaternion linear canonical transform (QLCT). Additionally, we discuss the rationale behind studying such transforms using quaternion functions as an illustrative example.

Bi-univalent functions associated with the leaf-like domain within the open unit disk are investigated and a new subclass is introduced and studied in the research presented here. This is achieved by applying the subordination principle for analytic functions in conjunction with q-calculus. The class is proved to be not empty. By proving its existence, generalizations can be given to other sets of functions. In addition, coefficient bounds are examined with a particular focus on |α2| and |α3| coefficients, and Fekete–Szegö inequalities are estimated for the functions in this new class. To support the conclusions, previous works are cited for confirmation.

Over the last few years, Geometric Function Theory (GFT) as one of the most prime branch of complex analysis has gained a considerable and an impressive attention from many researchers, largely because it deals with the study of the geometric properties of analytic functions and their numerous applications in various fields of mathematics such as in special functions, probability distributions, and fractional calculus. The investigations in this paper are on two new classes of analytic functions defined in the unit disk ε = {z ∈ C : |z| < 1} and denoted by XSq (b,K) and XTq (b,K). Function f in the classes satisfy the conditions f(0) = f ′(0) -- 1 = 0, hence can be of series type f(z) = z + a2z² + a3z³ + …, z ∈ ε. The definition of the two new classes of analytic functions embed some well-known special functions such as the Galuê-type Struve function, modified error function and a starlike function whose coefficients are Bell's numbers while some involving mathematical principles are the q-derivative, inequalities, convolution and subordination. The main results from these classes are however, the upper estimates of some initial bounds such as |an| (n = 2, 3, 4) and the Fekete-Szegö functional |a3 - øa2/2| (ø ∈ C) of functions f ∈ XSq (b,K) and f ∈ XTq (b,K). [ABSTRACT FROM AUTHOR]

This paper aims to study the Geometric properties of analytic function in the open unit disk. In the present investigation, we obtain some geometric properties of Pascal distribution involving Kamali type related to leaf like domain. In this paper, we find coefficient inequality, Radii Properties, convolution product, partial sum of the class $ \Sigma(\delta, \Phi, \beta, s, t, m) $. Furthermore, we examine the distortion bounds belonging to the same class.

In this paper, we introduce a convolution-type product for strongly integrable Hilbert $C^*$-module valued maps on locally compact groups. We investigate various properties of this product related to uniform continuity, boundless, etc. For instance, we prove a convolution theorem. Also, we study the boundless of the related convolution operator in various settings.

We introduce a new class of bi-univalent functions defined in the open unit disc and connected with a $q$-convolution. We find estimates for the general Taylor-Maclaurin coefficients of the functions in this class by using Faber polynomial expansions and we obtain an estimation for the Fekete-Szegö problem for this class.

The concept of subordination is expanded in this study from the fuzzy sets theory to the geometry theory of analytic functions with a single complex variable. This work aims to clarify fuzzy subordination as a notion and demonstrate its primary attributes. With this work’s assistance, new fuzzy differential subordinations will be presented. The first theorems lead to intriguing corollaries for specific aspects chosen to exhibit fuzzy best dominance. The work introduces a new integral operator for meromorphic functions and uses the newly developed integral operator, which is starlike and convex, respectively, to obtain conclusions on fuzzy differential subordination.

In this paper, we create a new subclass of convex functions given with tangent functions applying the combination of Babalola operators and Binomial series. Moreover, we obtain several important geometric results, including sharp coefficient bounds, sharp Fekete–Szego inequality, Kruskal inequality, and growth and distortion estimates. Furthermore, for functions with logarithmic coefficients, we compute sharp Fekete–Szego inequality and sharp coefficient bounds.

The focus of the present work is on the establishment and investigation of the coefficient estimates of two new subclasses of bi-close-to-convex functions and bi-concave functions; these are of an Ozaki type and involve a modified Caputo’s fractional operator that is associated with three-leaf functions in the open unit disc. The classes are defined using the notion of subordination based on the previously established fractional integral operators and classes of starlike functions associated with a three-leaf function. For functions in these classes, the Fekete-Szegö inequalities and the initial coefficients, |a2| and |a3|, are discussed. Several new implications of the findings are also highlighted as corollaries.

The quaternion windowed linear canonical transform is a tool for processing multidimensional data and enhancing the quality and efficiency of signal and image processing; however, it has disadvantages due to the noncommutativity of quaternion multiplication. In contrast, reduced biquaternions, as a special case of four-dimensional algebra, possess unique advantages in computation because they satisfy the multiplicative exchange rule. This paper proposes the reduced biquaternion windowed linear canonical transform (RBWLCT) by combining the reduced biquaternion signal and the windowed linear canonical transform that has computational efficiency thanks to the commutative property. Firstly, we introduce the concept of a RBWLCT, which can extract the time local features of an image and has the advantages of both time-frequency analysis and feature extraction; moreover, we also provide some fundamental properties. Secondly, we propose convolution and correlation operations for RBWLCT along with their corresponding generalized convolution, correlation, and product theorems. Thirdly, we present a fast algorithm for RBWLCT and analyze its computational complexity based on two dimensional Fourier transform (2D FTs). Finally, simulations and examples are provided to demonstrate that the proposed transform effectively captures the local RBWLCT-frequency components with enhanced degrees of freedom and exhibits significant concentrations.

The objective of the current examination is to present new sub-classes of q-convex and q-starlike functions inside E={z∈C:|z|

The first part of the paper describes a two-parameter (2P) fifth-order interpolation kernel, r. After that, from the 2P kernel, the kernel components were created. By applying the Fourier transformation to each kernel component, the spectral components of the 2P kernel were obtained. The spectral characteristic of the 2P kernel, H, was created from the spectral components. After that, the algorithm, that optimizes the parameters of the 2P kernel so as to eliminate the ripple of the spectral characteristics, is described. The optimization was performed in such a way that the spectral characteristic developed in the Taylor series, HT. With the condition for the elimination of the members of the Tylor series, which greatly affect the ripple of the spectral characteristic, the optimal kernel parameters (aopt, bopt) were determined. The second part of the paper describes an Experiment, in which the interpolation accuracy of the 2P kernel was tested. Convolution interpolation, with the 2P kernel, was performed over the signals from the Test base. The Test base is created with musical signals. By analyzing the interpolation error, which is represented by the Mean Square Error, MSE, the precision of the interpolation was determined. The results (aopt, bopt, MSEmin) are presented on tables and graphs. Detailed comparative analysis showed higher interpolation precision with the proposed 2P interpolation kernel, compared to the interpolation precision with, 1P interpolation kernel. Finally, the numerical values of the optimal kernel parameters, which are determined by the optimization algorithm proposed in this paper, were experimentally verified.

In this article, we investigate a class of analytic functions defined on the unit open disc U={z:∣z∣0\alpha \gt 0, 0≤β≤10\le \beta \le 1, 00{\rm{Re}}\left\{\frac{f^{\prime} \left(z)+\frac{1-\gamma }{\alpha \gamma }z{f}^{^{\prime\prime} }\left(z)-\beta }{1-\beta }\right\}\gt 0 holds. We find conditions on the numbers α,β\alpha ,\beta , and γ\gamma such that Pα(β,γ)⊆SP(λ){{\mathcal{P}}}_{\alpha }\left(\beta ,\gamma )\subseteq SP\left(\lambda ), for λ∈(−π2,π2)\lambda \in \left(-\frac{\pi }{2},\frac{\pi }{2}), where SP(λ)SP\left(\lambda ) denotes the set of all λ\lambda -spirallike functions. We also make use of Ruscheweyh’s duality theory to derive conditions on the numbers α,β,γ\alpha ,\beta ,\gamma and the real-valued function φ\varphi so that the integral operator Vφ(f){V}_{\varphi }(f) maps the set Pα(β,γ){{\mathcal{P}}}_{\alpha }\left(\beta ,\gamma ) into the set SP(λ)SP\left(\lambda ), provided φ\varphi is non-negative normalized function (∫01φ(t)dt=1)\left({\int }_{0}^{1}\varphi \left(t){\rm{d}}t=1) and Vφ(f)(z)=∫01φ(t)f(tz)tdt.{V}_{\varphi }(f)\left(z)=\underset{0}{\overset{1}{\int }}\varphi \left(t)\frac{f\left(tz)}{t}{\rm{d}}t.

In the present investigation, our main aim is to introduce a certain subclass of analytic univalent functions related to the Error function. We discuss the implications of our main results, including the coefficient bound, extreme points, weighted mean, convolution, convexity, and radii properties, as well as any other related properties.

Convolutional Neural Network (CNN) plays a vital role in the development of computer vision applications. The depth neural network composed of U-shaped structures and jump connections is widely used in various medical image tasks. Recently, based on the self-attention mechanism, the Transformer structure has made great progress and tends to replace CNN, and it has great advantages in understanding global information. In this paper, the ConvWin Transformer structure is proposed, which refers to the W-MSA structure in Swin and combines with the convolution. It can not only accelerate the convergence speed, but also enrich the information exchange between patches and improve the understanding of local information. Then, it is integrated with UNet, a U-shaped architecture commonly used in medical image segmentation, to form a structure called ConvWin-UNet. Meanwhile, this paper improves the patch expanding layer to perform the upsampling operation. The experimental results on the Hubmap datasets and synapse multi-organ segmentation dataset indicate that the proposed ConvWin-UNet structure achieves excellent results. Partial code and models of this work are available at https://github.com/xmFeng-hdu/ConvWin-UNet.

The paper extends an earlier result of G.V. Kalachev et al. (Sb. Math. 210(8):1129–1147, 2019) on the existence of a maximizer of convolution operator acting between two Lebesgue spaces on R n with kernel from some L q , 1 < q < ∞ . On the other hand, E. Lieb (Ann. of Math. 118:(2):349–374, 1983) proved the existence of a maximizer for the Hardy-Littlewood-Sobolev inequality and remarked that in general a convolution maximizer for a kernel from weak L q may not exist. In this paper we axiomatize some properties used in the proof of the Kalachev-Sadov 2019 theorem and obtain a more general result. As a consequence, we prove that the convolution maximizer always exists for kernels from a slightly more narrow class than weak L q , which contains all Lorentz spaces L q , s with q ≤ s < ∞ . [ABSTRACT FROM AUTHOR]

In the current exploration, we defined new subclasses of analytic functions, namely Rlim(l,ν) and Clim(l,ν), defined by subordination linked with a Limacon-shaped domain. We found a few initial coefficient bounds and Fekete–Szegő inequalities for the functions in the above-stated new classes. The corresponding results have been derived for the function h−1. Additionally, we discuss the Poisson distribution as an application of our consequences.

This article proposes asymptotic methods for calculating Sommerfeld integrals, which enable us to calculate the integral using the expansion of a function into an infinite power series at the saddle point, where the role of a rapidly oscillating function under the integral can be fulfilled either by an exponential or by its product by the Hankel function. The proposed types of Sommerfeld integrals are generalized on the basis of integral representations of the Hertz radiator fields in the form of the inverse Hankel transform with the subsequent replacement of the Bessel function by the Hankel function. It is shown that the numerical values of the saddle point are complex. During integration, reference or so-called standard integrals, which contain the main features of the integrand function, were used. As a demonstration of the accuracy of the technique, a previously known asymptotic formula for the Hankel functions was obtained in the form of an infinite series. The proposed method for calculating Sommerfeld integrals can be useful in solving the half-space Sommerfeld problem. The authors present an example in the form of an infinite series for the magnetic field of reflected waves, obtained directly through the Sommerfeld integral (SI).

Over the past ten years, analytical functions’ reputation in the literature and their application have grown. We study some practical issues pertaining to multivalent functions with bounded boundary rotation that associate with the combination of confluent hypergeometric functions and binomial series in this research. A novel subset of multivalent functions is established through the use of convolution products and specific inclusion properties are examined through the application of second order differential inequalities in the complex plane. Furthermore, for multivalent functions, we examined inclusion findings using Bernardi integral operators. Moreover, we will demonstrate how the class proposed in this study, in conjunction with the acquired results, generalizes other well-known (or recently discovered) works that are called out as exceptions in the literature.

Down syndrome (DS) or Trisomy 21 is a genetic disorder that causes intellectual and mental disability in fetuses. The most essential marker for detecting DS during the first trimester of pregnancy is nuchal translucency (NT). Effective segmentation of the NT contour from the ultrasound (US) images becomes challenging due to the presence of speckle noise and weak edges. This study presents a Convolutional Neural Network (CNN) based SegNet model using a Visual Geometry Group (VGG-16) for semantically segmenting the NT region from the US fetal images and providing a fast and affordable diagnosis during the early stages of gestation. A transfer learning approach using AlexNet is implemented to train the NT segmented regions for the identification of DS. The proposed model achieved a Jaccard index of 0.96 and classification accuracy of 91.7 %, sensitivity of 85.7 %, and a Receiver operating characteristic (ROC) of 0.95.

By considering the Bessel functions of the first kind of order ν and convolution structure, a new subclass of univalent holomorphic functions based of differential subordination is defined. Some geometric properties related to coefficient estimate, weighted mean, convolution preserving conditions and neighborhood concept are obtained.

Let 𝒜 p be the class of functions f of the form f(z)=zp+ap+1zp+1+ap+2zp+2+⋯f\left( z \right) = {z^p} + {a_{p + 1}}{z^{p + 1}} + {a_{p + 2}}{z^{p + 2}} + \cdots that are analytic in the open unit disk 𝕌. For f ∈ 𝒜 p, new integral and differential operator nf is considered. The object of the present paper is to discuss some interesting properties of nf and to consider some examples for our results.

The main goal of the paper concerns two parts, in the ˝rst one, we extend some results from Blot et al [5] under general hypothesis on the measure, in the second part, we introduce a new class of functions, which we call measure pseudo 𝒮 - asymptotically omega periodic functions. The obtained results is the extension of those established recently in the literature. Then we establish many interesting results on those functions, namely their characterization, we give also several properties of those class of functions as composition results, the invariance by translation, and the convolution product. All sections are illustrated by examples or counter examples.

The aim of this work is to discuss some conditions for Touchard polynomials to be in the classes TBb(ρ,σ) and TKb(ρ,σ). Also, we obtain some connection between Rη(D,E) and TKb(ρ,σ). Also, we investigate several mapping properties involving these subclasses. Further, we discuss the geometric properties of an integral operator related to the Touchard polynomial. Additionally, briefly mentioned are specific instances of our primary results. Also, several particular examples are presented.

In this paper, the harmonic function related to the q-Srivastava–Attiya operator is described to introduce a new class of complex harmonic functions that are orientation-preserving and univalent in the open-unit disk. We also cover some important aspects such as coefficient bounds, convolution conservation, and convexity constraints. Next, using sufficiency criteria, we calculate the sharp bounds of the real parts of the ratios of harmonic functions to their sequences of partial sums. In addition, for the first time some of the interesting implications of the q-Srivastava–Attiya operator in harmonic functions are also included.

We utilize quasi-subordination to analyze and introduce several new classes, and we construct a new operator by combining the error function and q-convolution. Additionally, we obtain estimates for the Fekete Szego functional and the Taylor–Maclaurin coefficients for functions in c2 and c3 new classes. Moreover, we discuss some applications of the operator.

The aim of this paper is to introduce and investigate some new subclasses of bi-concave functions using q-convolution and some applications. These special cases are obtaining by making use of a q- derivative linear operator. For the new introduced subclasses, the authors obtain the first two initial Taylor–Maclaurin coefficients |c2| and |c3| of bi-concave functions. For certain values of the parameters, the authors deduce interesting corollaries for coefficient bounds which imply special cases of the new introduced operator. Also, we develop two examples for coefficients |c2| and |c3| for certain functions.

In the current article, we utilize the concept of subordination to establish a new subclass of analytic functions associated with a bounded domain that is symmetric about the real axis. By applying the convolution technique, we derive the necessary and sufficient condition, the radius of convexity for this recently introduced class. Furthermore, we prove the sharp upper bounds for the second-order Hankel determinants |H2,1ξ|,|H2,2ξ| and third-order Hankel determinant |H3,1ξ| for the functions ξ belonging to the newly defined class.

The Theory of Complex Functions has been studied by many scientists and its application area has become a very wide subject. Harmonic functions play a crucial role in various fields of mathematics, physics, engineering, and other scientific disciplines. Of course, the main reason for maintaining this popularity is that it has an interdisciplinary field of application. This makes this subject important not only for those who work in pure mathematics, but also in fields with a deep-rooted history, such as engineering, physics, and software development. In this study, we will examine a subclass of Harmonic functions in the Theory of Geometric Functions. We will give some definitions necessary for this. Then, we will define a new subclass of complex-valued harmonic functions, and their coefficient relations, growth estimates, radius of univalency, radius of starlikeness and radius of convexity of this class are investigated. In addition, it is shown that this class is closed under convolution of its members.