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Motivated by applications in Bayesian analysis, we introduce a multidimensional beta distribution in the ordered simplex. We study the properties of this distribution and connect them with the generalized incomplete beta function. This function is crucial in applications of multidimensional beta distribution. Thus, we present two efficient numerical algorithms for computing the generalized incomplete beta function, one based on Taylor series expansion and another based on Chebyshev polynomials. [ABSTRACT FROM AUTHOR]

In this article, we aim to present a new extension of the beta-logarithmic function with the help of an extended beta function and logarithmic mean. Then we introduce its statistical properties and investigate some basic important formulas of extended beta-logarithmic functions, such as integral representation, summation formula and Mellin transform. Further, we consider a new type of hypergeometric and confluent hypergeometric functions using this extended beta-logarithmic function and discuss their various result. [ABSTRACT FROM AUTHOR]

Abstract Recently, extensions of the gamma and beta functions have been studied due to their appealing properties and wide range of applications in various scientific fields. This note aims to investigate generalized inequalities associated with the extended beta and gamma functions.

In this our second article on the fourth Lauricella function, we start with some lemmas where Lauricella functions with m equal variables or with m copies of 1 can be reduced to a similar function. In the same way, Lauricella functions with m parameters equal to - 1 can be reduced to a sum of Lauricella functions times elementary symmetric polynomials of the variables. These formulas are used in the proofs, as well as transformation formulas for Lauricella functions from the first paper. Our method gives a transformation with Kampé de Fériet functions, which could possibly be extended. Also summation formulas for the first Appell function are proved. Because of the symmetric proofs, some q-analogues of these formulas can be found at the end of paper. All proofs use Eulerian (q-)integrals. Several formulas are generalizations of Kummer's second summation formula. In particular, the Euler–Pfaff transformation formula can be generalized to Lauricella functions. The sections are ordered according to the respective substitutions, and the recurring theme is again the reduced roots, which turn up as variables in the formulas. The power substitutions lead to formulas with complex function arguments, only one example is given. [ABSTRACT FROM AUTHOR]

Recently, extensions of the gamma and beta functions have been studied due to their appealing properties and wide range of applications in various scientific fields. This note aims to investigate generalized inequalities associated with the extended beta and gamma functions. [ABSTRACT FROM AUTHOR]

The Kullback–Leibler divergence (KL divergence) is a statistical measure that quantifies the difference between two probability distributions. Specifically, it assesses the amount of information that is lost when one distribution is used to approximate another. This concept is crucial in various fields, including information theory, statistics, and machine learning, as it helps in understanding how well a model represents the underlying data. In a recent study by Nawa and Nadarajah, a comprehensive collection of exact expressions for the Kullback–Leibler divergence was derived for both multivariate and matrix-variate distributions. This work is significant as it expands on our existing knowledge of KL divergence by providing precise formulations for over sixty univariate distributions. The authors also ensured the accuracy of these expressions through numerical checks, which adds a layer of validation to their findings. The derived expressions incorporate various special functions, highlighting the mathematical complexity and richness of the topic. This research contributes to a deeper understanding of KL divergence and its applications in statistical analysis and modeling. [ABSTRACT FROM AUTHOR]

The main aim of the present paper is to apply the extended Riemann Liouville fractional derivative operator for finding some linear generating relations for the Hermite polynomials. Four main results are obtained, which are presented in the form of four theorems. [ABSTRACT FROM AUTHOR]

By using the weight coefficients and the techniques of real analysis, a new extended Mulholland’s inequality with multi-parameters involving one partial sum is given. The equivalent statements of the best value related to several parameters are provided. The equivalent forms, some inequalities in particular parameters, and the operator expressions are obtained.

In this investigation, we propose a numerical method based on the fractional‐order generalized Taylor wavelets (FGTW) for option pricing and the fractional Black–Scholes equations. This model studies option pricing when the underlying asset has subdiffusive dynamics. By applying the regularized beta function, we give an exact formula for the Riemann–Liouville fractional integral operator (RLFIO) of the FGTW. An error analysis of the numerical scheme for estimating solutions is performed. Finally, we conduct a variety of numerical experiments for several standard examples from the literature to assess the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two one-parameter families of quadratic polynomial enrichments designed to enhance the accuracy of the classical Crouzeix–Raviart finite element. These enrichments are realized by using weighted line integrals as enriched linear functionals and quadratic polynomial functions as enrichment functions. To validate the effectiveness of our approach, we conduct numerical experiments that confirm the improvement achieved by the proposed method. [ABSTRACT FROM AUTHOR]

The Kullback–Leibler divergence is a measure of the divergence between two probability distributions, often used in statistics and information theory. However, exact expressions for it are not known for multivariate or matrix-variate distributions apart from a few cases. In this paper, exact expressions for the Kullback–Leibler divergence are derived for over twenty multivariate and matrix-variate distributions. The expressions involve various special functions. [ABSTRACT FROM AUTHOR]

The generalizations of hypergeometric and confluent hypergeometric functions, as well as gamma and beta functions, are the subject of numerous studies. The product of two Wright functions is used in this paper to define a new type of generalized beta function. Confluent hypergeometric functions and new types of generalized Gauss functions are obtained with the aid of the generalized beta function. Additionally, certain characteristics of these functions are established, including transform formulas, Mellin transforms, derivative formulas, integral representations, and summation formulas. [ABSTRACT FROM AUTHOR]

The beta (type 1) and inverted beta distributions are of vital importance in science, engineering and economics. In this article, we introduce a matrix variate generalization of the inverted beta distribution. By using linear and quadratic transformations, we obtain several matrix variate distributions including matrix variate generalized beta type 1 distribution. [ABSTRACT FROM AUTHOR]

Deep neural network-based machine learning algorithms are widely used within different sectors and produce excellent results. However, their use requires access to private, often confidential, and sensitive information (financial, medical, etc). This requires precise measures and particular attention to data security and confidentiality. In this paper, we propose a new solution to this problem by integrating a proposed Convolutional Neural Network (CNN) model on encrypted data within the constraints of homomorphic encryption techniques. Specifically, we focus on the approximate activation functions ReLU, Sigmoid, and Tanh, which appear to be the key functions of CNNs. We start by developing new low-degree polynomials, which are essential for successful Homomorphic Encryption (HE). The activation functions will be replaced by these polynomials, which are based on the Beta function and its primitive. To make certain that the data is contained within a given range, the next step is to build a new CNN model using batch normalization. Finally, our methodology and the effectiveness of the proposed strategy are evaluated using Mnist and Cifar10. The experimental results support the proposed approach's efficiency. [ABSTRACT FROM AUTHOR]

Copyright of Journal of South China Normal University (Natural Science Edition) / Huanan Shifan Daxue Xuebao (Ziran Kexue Ban) is the property of Journal of South China Normal University (Natural Science Edition) Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we introduce new transformations for Horn's double hypergeometric functions G1,G2 and G3. Also deduce new integral representations of Euler-type involving these functions in terms of hypergeometric functions of two variables. [ABSTRACT FROM AUTHOR]

The objective of this paper is to explore certain properties involving the generalized hypergeometric function ..., as introduced by Khan et al. [27]. Several fractional derivatives and integral formulas involving the generalized hypergeometric function are computed. Further, we derive the solution of generalized fractional kinetic equations involving generalized hypergeometric function. The respective solution is given in terms of Mittag-Leffler function. [ABSTRACT FROM AUTHOR]

Extended Abstract Background: Sustainable management and fundamental planning of forest resources require efficient and high-quality information about the current and future states of forests. Investigating the current and future states of forest stands, describing the structure of the forest stand, and the response of the stand to tending operations are among the capabilities of statistical distributions in forest management. Frequency distribution modeling is the distribution of people in a population in different classes. Since natural phenomena include continuous and discrete types, their distribution will also be different and diverse. The use of statistical distributions in the forest has a long history, and some researchers have considered its history as long as two centuries. Zagros forests, as one of the most important vegetation areas of Iran, have a special importance from the socioeconomic and ecological points of view. The dominant species of these forests is the Iranian oak (Quercus brantii Lindl). According to geographical and environmental conditions, this species has various habitats in the vegetation zone of the middle Zagros. It is expected that this species has different probability distribution functions in different habitats. Therefore, the probability distribution functions of the characteristic height in habitats with different vegetative structures of Iranian oak in the forests of Khorramabad city were investigated in this research to select the best models in different habitats. This study also seeks to provide useful information on the authority of forest managers and planners to preserve and improve these valuable forests. Methods: To carry out this research, two areas with different structures, including coppice with standard (Sefidkoh area) and standard with coppice (Qala-Gol area) were determined in the forests of Khorram Abad city through numerous excursions in the forest. A forest stand with an area of approximately 3 ha was selected in each of these areas. This forest stand was chosen in such a way to represent the forests of the region. The subsoil was semi-deep and north direction in the Qala-Gol forest stand at an altitude of 1850 m above sea level and with an average slope of 20%. In the SefidKoh forest stand, the subsoil was semi-deep and located in the north direction at an altitude of 1900 m from sea level with an average slope of 30%. In each of these two selected forest stands, the height of all Iranian oak trees, with a height of > 1.5 m, was counted 100%. In total, the heights of 458 and 348 Iranian oak trees were measured in the two SefidKoh and Qala-Gol regions, respectively. A TruPulse 360 laser device was used to measure the height of trees. To investigate the distribution of height classes, the most widely used continuous probability distribution models in biological research were used due to the continuousness of the height variable, including two-parameter Weibull, two-parameter gamma, normal, log-normal, beta, and Johnson. The goodness of fit of the data was also examined using the Anderson-Darling test, which has a higher power than other tests. Easy Fit 5.6 software was used to analyze all data and draw graphs. Results: The statistical errors were 4.45% and 2.82% in the Sefidkoh and Qala-Gol regions, respectively. The average height of trees was 6.76 m in the Sefidkoh region (between 1.8 and 18 m) compared to that of 7.54 m (between 1.53 and 14.04 m) measured in the Qala-Gol area. The values of skewness and Kurtosis were positive in the two forest areas, with higher values in the Sefidkouh region than in the Qala-Gol area, indicating the concentration of data in lower-height classes in the Sefidkouh region. The results of Anderson-Darling's goodness of fit test in the Sefidkoh area (coppice with standard) showed that the most suitable probability distribution functions (with a 5% error level) for modeling the distribution of trees in height classes were Johnson sb and beta, respectively. In the Qala-Gol region (standard with coppice), the most appropriate distribution functions were identified as beta and normal, respectively. Conclusion: The results of this research show that the forest has an uneven-aged structure in the areas where the oak forest has a coppice with standard form, according to the form of distribution (decreasing distribution). In contrast, the forest has moved toward a single-stemmed and even-aged forest in areas where the forest form is standard with coppice. This is mainly caused by the destruction through grazing, the cutting of branches for livestock and fuel consumption, and the indiscriminate use of byproducts that have occurred in this region. In the Sefidkoh region, the effects of destruction and less human intervention were observed at the regional level in field observations. According to the shape of the distributions obtained in the two sites, the vegetative forms of the trees affected the distribution functions of their heights, which can provide useful information for managers, planners, and researchers to maintain and control these valuable forests.

Two generalized integrals of the beta family are the prime focus of this paper. By taking into account the generalized integral of the beta family, the series and integral representations are created through generalized special functions. Also covered are the fractional maps of Saigo, Riemann–Liouville, and Kober operators with the extended beta function. Results for classical beta function and extended beta functions were proved as special cases. [ABSTRACT FROM AUTHOR]

The primary object of this article is to introduce (p, q)-beta logarithmic function with extended beta function by using the logarithmic mean. We evaluate different properties and representations of beta logarithmic function. Further, it is evaluated logarithmic distribution, hypergeometric and confluent hypergeometric functions via logarithmic mean are evaluated and their essential properties are studied. Numerous formulas of (p, q)-beta logarithmic functions such as integral formula, derivative formula, transformation formula and generating function are analyzed. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to introduce a new extension of the generalized Hurwitz-Lerch Zeta functions of two variables [abbreviated as H-L-Z-F-2V] and three variables [abbreviated as H-L-ZF-3V] involving the extended Beta function. We then systematically investigate its several interesting properties and related formulas as (for example) various integral representations, which provide certain new and known extensions of earlier corresponding results, a derivative formula, and Mellin-transform. We also consider some important special cases [ABSTRACT FROM AUTHOR]

In this paper we apply the Riemann–Liouville, Erdelyi–Kober and Caputo fractional operators to the modified beta, modified Gauss hypergeometric and modified confluent hypergeometric functions in which the generalized M-series are included in their kernels. Furthermore, as examples, we obtain solutions of some fractional differential equations involving the above modified special functions.

The generalized $M$-series is a hybrid function of generalized Mittag-Leffler function and generalized hypergeometric function. The principal aim of this paper is to investigate certain properties resembling those of the Mittag-Leffler and Hypergeometric functions including various differential and integral formulas associated with generalized $M$-series. Certain corollaries involving the generalized hypergeometric function are also discussed. Further, in view of Hadamard product of two analytic functions, we have represented our main findings in Hadamard product of two known functions.

This study performed a numerical investigation to explore the effect of span-wise area distribution on flapping wing aerodynamics in hovering configuration using an in-house developed flow solver. The span-wise area distribution was defined using a morphological beta function and the flapping motion was set to a sinusoidal movement manner. Results show that moving the area distribution toward the tip region can generate more lift and simultaneously expense more power, whereas an optimum area distribution (${\rm{\bar{r}}}_1$ = 0.45) was observed because of its aerodynamic efficiency. In general, the temporal profiles of aerodynamic forces are slightly sensitive concerning span-wise area distribution rather than the peak and mean force. Detail flow structure visualization illustrated that the flapping wing locomotion produces complex spatial and temporal vortex structures, including vortex generation, development, and shedding of leading-edge vortex, trailing edge vortex, and root vortex. For flapping wing with a larger area on the tip is in principle capable of enhancing the vortex strength, particularly for the leading-edge vortex which dominates the lift generation during flapping motion. Meanwhile, the smoother profile bounded by the tip and leading edge is beneficial for stabilizing the leading edge and tip vortex.

Machine learning algorithms based on deep neural networks have been widely used in many fields especially in computer vision, with impressive results. However, these models are vulnerable to different types of attacks like adversarial ones, which require attention to the model security and confidentiality. This study proposes a defense strategy to improve the insurance of white-box models while minimizing adversarial attacks against Fast Gradient Sign Method (FGSM)-based attacks and tackling the issue of adversarial training to improve their robustness. Mainly, we proposed a CNN model based on personalized activation functions using Beta function and its primitive. Then, the new resulted low degree polynomials are used to approximate the ReLU, Sigmoid and Tanh activation functions. Batch normalization was evoked to significantly improve the learning capacity of neural networks. The obtained results, using Mnist dataset prove the effectiveness of the proposed model.

In this article, we present new fractional integral inequalities via Euler’s beta function in terms of ss-convex mappings. We develop some new generalizations of fractional trapezoid- and midpoint-type inequalities using the class of differentiable ss-convexity. The results obtained in this study extended other related results reported in the literature.

A collection of functions organized according to their indexing based on non-negative integers is grouped by the common factor of fixed integer N. This grouping results in a summation of N series, each consisting of functions partitioned according to this modulo N rule. Notably, when N is equal to two, the functions in the series are divided into two subseries: one containing even-indexed functions and the other containing odd-indexed functions. This partitioning technique is widely utilized in the mathematical literature and finds applications in various contexts, such as in the theory of hypergeometric series. In this paper, we employ this partitioning technique to establish four distinct families of summation formulas for F 3 4 (1) hypergeometric series. Subsequently, we leverage these summation formulas to introduce eight categories of integral formulas. These integrals feature compositions of Beta function-type integrands and F 2 3 (x) hypergeometric functions. Additionally, we highlight that our primary summation formulas can be used to derive some well-known summation results. [ABSTRACT FROM AUTHOR]

In this paper, we define generalized Fourier and inverse Fourier transforms containing the h-exponential function in their kernels and give the fundamental properties of these transforms. We also compute the transforms of both the classical and the generalized Riemann-Liouville and Caputo fractional operators. In addition, we compute the transforms of some elementary and generalized special functions as well. Finally, as applications, we obtain the solutions of two differential equations with ordinary and fractional derivatives using the transforms we have defined. [ABSTRACT FROM AUTHOR]

We consider the following generalization of the classical coupon collector problem. We assume that, in addition to the initial collection of standard coupons, there is one more coupon that acts as a reset button, removing all coupons from the part of the collection that has already been drawn. For the case where standard coupons have unequal probabilities of being drawn, we obtain the distribution of the waiting time until the end of the collection process. For the case where standard coupons have equal probabilities, we derive a simple formula for the expected waiting time in terms of the beta function, and discuss the asymptotic properties of this expected waiting time, when the number of standard coupons tends toward infinity. [ABSTRACT FROM AUTHOR]

In this research article, we provide an extended Hurwitz–Lerch zeta function using the extended beta function. We have also investigated the new generalized Hurwitz–Lerch zeta function's exciting characteristics, such as integral representations, differential equations, the Mellin transform and special generating relations. In order to support the main result, we have additionally considered several notable specific cases. Furthermore, we have discussed the solution of the fractional kinetic equation using the new extended Hurwitz–Lerch zeta function. [ABSTRACT FROM AUTHOR]

The article focuses on exploring relations involving special functions like the gamma, beta, and zeta functions. Topics including their connections to integrals, infinite sums, inequalities and specific value calculations. It delves into the historical context and applications of special functions in mathematics, physics and engineering; and highlighting their diverse nature and importance in various mathematical disciplines.

Background: Sustainable management and fundamental planning of forest resources require efficient and high-quality information about the current and future states of forests. Investigating the current and future states of forest stands, describing the structure of the forest stand, and the response of the stand to tending operations are among the capabilities of statistical distributions in forest management. Frequency distribution modeling is the distribution of people in a population in different classes. Since natural phenomena include continuous and discrete types, their distribution will also be different and diverse. The use of statistical distributions in the forest has a long history, and some researchers have considered its history as long as two centuries. Zagros forests, as one of the most important vegetation areas of Iran, have a special importance from the socioeconomic and ecological points of view. The dominant species of these forests is the Iranian oak (Quercus brantii Lindl). According to geographical and environmental conditions, this species has various habitats in the vegetation zone of the middle Zagros. It is expected that this species has different probability distribution functions in different habitats. Therefore, the probability distribution functions of the characteristic height in habitats with different vegetative structures of Iranian oak in the forests of Khorramabad city were investigated in this research to select the best models in different habitats. This study also seeks to provide useful information on the authority of forest managers and planners to preserve and improve these valuable forests. Methods: To carry out this research, two areas with different structures, including coppice with standard (Sefidkoh area) and standard with coppice (Qala-Gol area) were determined in the forests of Khorram Abad city through numerous excursions in the forest. A forest stand with an area of approximately 3 ha was selected in each of these areas. This forest stand was chosen in such a way to represent the forests of the region. The subsoil was semi-deep and north direction in the Qala-Gol forest stand at an altitude of 1850 m above sea level and with an average slope of 20%. In the SefidKoh forest stand, the subsoil was semi-deep and located in the north direction at an altitude of 1900 m from sea level with an average slope of 30%. In each of these two selected forest stands, the height of all Iranian oak trees, with a height of > 1.5 m, was counted 100%. In total, the heights of 458 and 348 Iranian oak trees were measured in the two SefidKoh and Qala- Gol regions, respectively. A TruPulse 360 laser device was used to measure the height of trees. To investigate the distribution of height classes, the most widely used continuous probability distribution models in biological research were used due to the continuousness of the height variable, including two-parameter Weibull, two-parameter gamma, normal, log-normal, beta, and Johnson. The goodness of fit of the data was also examined using the Anderson-Darling test, which has a higher power than other tests. Easy Fit 5.6 software was used to analyze all data and draw graphs. Results: The statistical errors were 4.45% and 2.82% in the Sefidkoh and Qala-Gol regions, respectively. The average height of trees was 6.76 m in the Sefidkoh region (between 1.8 and 18 m) compared to that of 7.54 m (between 1.53 and 14.04 m) measured in the Qala-Gol area. The values of skewness and Kurtosis were positive in the two forest areas, with higher values in the Sefidkouh region than in the Qala-Gol area, indicating the concentration of data in lower-height classes in the Sefidkouh region. The results of Anderson-Darling's goodness of fit test in the Sefidkoh area (coppice with standard) showed that the most suitable probability distribution functions (with a 5% error level) for modeling the distribution of trees in height classes were Johnson sb and beta, respectively. In the Qala-Gol region (standard with coppice), the most appropriate distribution functions were identified as beta and normal, respectively. Conclusion: The results of this research show that the forest has an uneven-aged structure in the areas where the oak forest has a coppice with standard form, according to the form of distribution (decreasing distribution). In contrast, the forest has moved toward a single-stemmed and evenaged forest in areas where the forest form is standard with coppice. This is mainly caused by the destruction through grazing, the cutting of branches for livestock and fuel consumption, and the indiscriminate use of byproducts that have occurred in this region. In the Sefidkoh region, the effects of destruction and less human intervention were observed at the regional level in field observations. According to the shape of the distributions obtained in the two sites, the vegetative forms of the trees affected the distribution functions of their heights, which can provide useful information for managers, planners, and researchers to maintain and control these valuable forests. [ABSTRACT FROM AUTHOR]

In this paper, we have defined new generalizations of some hypergeometric functions and fractional operators with the help of Fox-Wright function. Then, using each of the generalized fractional operators, we derived linear and bilinear generating function relations for these functions. Finally, we have shown that the newly defined hypergeometric functions and fractional operators can be reduced to functions and operators presented in many studies in the literature by giving special values for their parameters. [ABSTRACT FROM AUTHOR]

The Beta-logarithmic function, providing simultaneously a generalization of the logarithmic mean and the beta function, has been recently introduced by the authors. In this paper we deal with approximating this function when investing some advanced integral inequalities. Many old/new inequalities involving the standard beta function and the logarithmic mean are immediately deduced. [ABSTRACT FROM AUTHOR]

In this paper, we introduce three types of generalized special functions: beta, Gauss hypergeometric, and confiuent hypergeometric, all involving two generalized M-series at their kernels. We then give several properties of these functions, such as integral representations, functional relations, summation relations, derivative formulas, transformation formulas, and double Laplace transforms. Furthermore, we obtain solutions of fractional partial differential equations involving these new generalized special functions and then we present graphs of the approximate behavior of the solutions. Also, we introduce a new generalized beta distribution and incomplete beta function. Finally, we establish relationships between the new generalized special functions and other generalized special functions found in the literature. [ABSTRACT FROM AUTHOR]

Recently, Kim-Kim introduced the truncated degenerate Bell polynomials and numbers. In this paper, weintroduce the truncated Lah-Bell polynomials and numbers. We obtain some identities, recurrence relations and properties. Furthermore, we also introduce the truncated modified Lah-Bell polynomials and numbers, truncated degenerate Lah-Bell polynomials and numbers and we derive some properties and identities involving with those polynomials and numbers. [ABSTRACT FROM AUTHOR]

In this article, generalized pathway integral operator with the classical Gauss hypergeometric function kernel and the fractional differential operators are used to studied new extended hypergeometric function. Furthermore, extended fractional kinetic equation involving new extended hypergeometric function that contained two Fox-Wright function as regularizer is investigated using Sumudu transform and Wiman's function. [ABSTRACT FROM AUTHOR]

In the present article, our main goal is to study the new extended beta, Gauss and confluent hypergeometric functions defined by using the two Fox-Wright functions as their kernel. Some applications of the new extended beta function to distribution theory are also discussed. [ABSTRACT FROM AUTHOR]

In this work, by constructing a new kernel function in general form which includes both the homogeneous and the non-homogeneous cases, a half-discrete Hilbert-type inequality involving the newly constructed kernel function is established. Additionally, the equivalent Hardy-type inequalities are considered, and all the constant factors in the newly obtained inequalities are proved to be the best possible. Furthermore, by specializing the kernel function and introducing some special functions such as Beta function and Gamma function, some existing results and new examples are presented at the end of the paper. [ABSTRACT FROM AUTHOR]

In this paper, we establish some theorems in order to evaluate certain single and double integral formulas associated with the generalized M-series. The results are generalizations and unification of formulas for a fairly general class of special functions with a focus on the Mittag-Leffler function and hypergeometric function. By specializing the parameters, some new or known integrals are also investigated as corollaries in terms of generalized hypergeometric function. Further, applying the Hadamard product of two analytic functions, we have represented all our main findings in the Hadamard product of two known functions. [ABSTRACT FROM AUTHOR]

Abstract By means of the weight functions, the idea of introduced parameters, using the transfer formula and Hermite–Hadamard’s inequality, a more accurate half-discrete multidimensional Hilbert-type inequality with the homogeneous kernel as 1 ( x + ∥ k − ξ ∥ α ) λ + m ( x , λ > 0 ) $\frac{1}{(x + \Vert k - \xi \Vert _{\alpha} )^{\lambda + m}}\ (x,\lambda > 0)$ involving one derivative function of m-order is given. The equivalent conditions of the best possible constant factor related to several parameters are considered. The equivalent forms. the operator expressions and some particular inequalities are obtained.

Abstract By means of the weight functions, the idea of introducing parameters and the technique of real analysis related to the beta and gamma functions, a new reverse Hardy–Hilbert-type integral inequality with the homogeneous kernel as 1 ( x + y ) λ + m + n $\frac{1}{(x + y)^{\lambda + m + n}}$ ( λ > 0 $\lambda > 0$ ) involving two derivative functions of higher order is given. As applications, the equivalent statements of the best possible constant factor related to several parameters are considered, and some particular inequalities are obtained.

Abstract By means of the weight functions, the idea of introducing parameters and the technique of real analysis, a new Hardy–Hilbert-type integral inequality with the homogeneous kernel 1 ( x + y ) λ ( λ > 0 ) $\frac{1}{(x + y)^{\lambda}}\ (\lambda > 0)$ involving two multiple upper-limit functions is obtained. The equivalent statements of the best possible constant factor related to the beta and gamma functions are considered. As applications, the equivalent forms and the case of a nonhomogeneous kernel are deduced. Some particular inequalities and the operator expressions are provided.