922 Results Found

In the study, the authors introduce Qi’s normalized remainder of the Maclaurin power series expansion of the function $\ln \sec x=-\ln \cos x$; in view of a monotonicity rule for the ratio of two Maclaurin power series and by virtue of the logarith...

The beta-logarithmic function substantially generalizes the standard beta function, which is widely recognized for its significance in many applications. This article is devoted to the study of a generalization of the classical beta-logarithmic funct...

The logarithmic coefficients play an important role for different estimates in the theory of univalent functions. Due to the significance of the recent studies about the logarithmic coefficients, the problem of obtaining the sharp bounds for the modu...

The logarithmic functions have been used in a verity of areas of mathematics and other sciences. As far as we know, no one has used the coefficients of logarithmic functions to determine the bounds for the third Hankel determinant. In our present inv...

Using the Lebedev–Milin inequalities, bounds on the logarithmic coefficients of an analytic function can be transferred to estimates on coefficients of the function itself and related functions. From this fact, the study of logarithmic-related...

This is a collection of definite integrals involving the logarithmic and polynomial functions in terms of special functions and fundamental constants. All the results in this work are new.

In our present study, two subclasses of starlike functions which are symmetric about the origin are considered. These two classes are defined with the use of the sigmoid function and the trigonometric function, respectively. We estimate the first fou...

In this methodological–technical note, in addition to the well-known concepts of logarithms of positive real numbers and operators, we open a path for mathematical treatment of the mathematical concept of the logarithm of a vector. We prove the...

In view of a general formula for higher order derivatives of the ratio of two differentiable functions, the authors establish the first form for the Maclaurin power series expansion of a logarithmic expression in term of determinants of special Hesse...

The concept of coefficient estimates on univalent functions is one of the interesting aspects explored recently by many researchers. Motivated by this direction, in this present work, we obtain the upper bounds of initial inverse coefficients and log...

A class of definite integrals involving a quotient function with a reducible polynomial, logarithm and nested logarithm functions are derived with a possible connection to contact problems for a wedge. The derivations are expressed in terms of the Le...

We consider a Keller–Segel type chemotaxis model with logarithmic sensitivity and logistic growth. The logarithmic singularity in the system is removed via the inverse Hopf–Cole transformation. We then linearize the system around a constant equilibri...

In this paper, first derivatives of the Whittaker function ${\mathrm{M}}_{\kappa ,\mu }\left(x\right)$ are calculated with respect to the parameters. Using the confluent hypergeometric function, these derivarives can be expressed as infinite sums of quotients of the digamma and...

The purpose of this article is to obtain the sharp estimates of the first four initial logarithmic coefficients for the class ${\mathcal{BT}}_s$ of bounded turning functions associated with a petal-shaped domain. Further, we investigate the sharp estimate of Fekete...

Generalized numbers, arithmetic operators, and derivative operators, grouped in four classes based on symmetry features, are introduced. Their building element is the pair of q-logarithm/q-exponential inverse functions. Some of the objects were previ...

Recently, we have established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of analytical function to obtain new criteria equivalent to the Riemann hypothesis. Later, the same theorem was applied to calcula...

Recently, we established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of analytical function to obtain new criteria equivalent to the Riemann hypothesis. The same theorem was subsequently applied to calcul...

Recently, we have established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of an analytical function to obtain a few new criteria equivalent to the Riemann hypothesis. Here, the same theorem is applied to...

The study explores analytic, geometric and algebaraic properties of two subclasses of analytic functions: the class of Bounded Radius Rotation denoted by ${\mathcal{R}}_{\mathfrak{s}}{,}_{\varrho }(\mathcal{A},\mathcal{B},z)$, and the class of Bounded Boundary Rotation denoted by ${\mathcal{V}}_{\mathfrak{s}}{,}_{\varrho }(\mathcal{A},\mathcal{B},z)$, both...

Determining the optimal portfolio is important in the investment process because it includes the selection of appropriate fund allocation to manage financial risk effectively. Although risk cannot be entirely eliminated, it is managed through strateg...

The purpose of this study was to obtain the sharp Hankel determinant $H_{2,1}\left(F_f/2\right)$ and $H_{2,2}\left(F_f/2\right)$ with a logarithmic coefficient as entry for the class ${\mathcal{BT}}_{3\mathcal{L}}$ of bounded turning functions connected with a three-leaf-shaped domain. In this study, we developed...

Seismic fragility analysis is an efficient method to evaluate the structural failure probability during earthquake events. Among the existing fragility analysis methods, the probabilistic seismic demand model (PSDM) and the joint probabilistic seismi...

Memristors are state-of-the-art, nano-sized, two-terminal, passive electronic elements with very good switching and memory characteristics. Owing to their very low power usage and a good compatibility to the existing CMOS ultra-high-density integrate...

In this paper, we have derived and evaluated a quadruple integral whose kernel involves the logarithm and product of Bessel functions of the first kind. A new quadruple integral representation of Catalan’s G and Apéry’s $\zeta \left(3\right)$ c...

Technology includes hard technology and soft technology. Material technology embodied in production conditions and working conditions such as machinery, equipment and infrastructure is called hard technology, referring to the technology directly used...

In this paper, a new subclass has been defined as $\Omega$ of the univalent function in $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$. The central goal of this paper is to determine estimates for logarithmic coefficients, inverse logarithmic coefficients, some cases of the H...

The derivation of integrals in the table of Gradshteyn and Ryzhik in terms of closed form solutions is always of interest. We evaluate several of these definite integrals of the form ${\int }_0^{\infty }log(1\pm e^{-\alpha y})$...

In this work, the bounds for the logarithmic coefficients ${\gamma }_n$ of the general classes ${\mathcal{S}}^{*}\left(\phi \right)$ and $\mathcal{K}\left(\phi \right)$ were estimated. It is worthwhile mentioning that the given bounds would generalize some of the previo...

We present a method using contour integration to evaluate the definite integral of the form ${\int }_0^{\infty }{log}^k\left(ay\right)R\left(y\right)dy$ in terms of special functions, where $R\left(y\right)=\frac{y^m}{1+\alpha y^n}$ and $k,m$...

While traditional support vector regression (SVR) models rely on loss functions tailored to specific noise distributions, this research explores an alternative approach: $\epsilon$-ln SVR, which uses a loss function based on the natural logarithm of t...

Starlike and convex functions have gained increased prominence in both academic literature and practical applications over the past decade. Concurrently, logarithmic coefficients play a pivotal role in estimating diverse properties within the realm o...

In the paper, by virtue of a derivative formula for the ratio of two differentiable functions and with the help of a monotonicity rule, the authors expand a logarithmic expression involving the sine function into the Maclaurin power series in terms o...

In this paper, we obtain the sharp and accurate bounds for the logarithmic coefficients of some subclasses of analytic functions defined and studied in earlier works. Furthermore, we obtain the bounds of the second Hankel determinant of logarithmic c...

In the paper, (1) in view of a general formula for any derivative of the quotient of two differentiable functions, (2) with the aid of a monotonicity rule for the quotient of two power series, (3) in light of the logarithmic convexity of an elementar...

In this paper, exact solutions of semilinear equations having exponential growth in the space variable x are found. Semilinear Schrödinger equation with logarithmic nonlinearity and third-order evolution equations arising in optics with logarith...

A novel approach to constructing an algorithm for computing discrete logarithms, which holds significant interest for advancing cryptographic methods and the applied use of multivalued logic, is proposed. The method is based on the algebraic delta fu...

In this paper, function approximation is utilized to establish functional series approximations to integrals. The starting point is the definition of a dual Taylor series, which is a natural extension of a Taylor series, and spline based series appro...

Using the Dunford–Taylor integral and a representation formula for the resolvent of a non-singular complex matrix, we find the logarithm of a non-singular complex matrix applying the Cauchy’s residue theorem if the matrix eigenvalues are...

A quadruple integral involving the logarithmic, exponential and polynomial functions is derived in terms of the Lerch function. Special cases of this integral are evaluated in terms of special functions and fundamental constants. Almost all Lerch fun...

In our present investigation, some coefficient functionals for a subclass relating to starlike functions connected with three-leaf mappings were considered. Sharp coefficient estimates for the first four initial coefficients of the functions of this...

In numerous geometric and physical applications of complex analysis, estimating the sharp bounds of coefficient-related problems of univalent functions is very important due to the fact that these coefficients describe the core inherent properties of...

In this research article, we introduce new family ${\mathcal{R}}_G$ of holomorphic functions, which is related to the generalized bounded turning and generating functions of Gregory coefficients. Leveraging the concept of functions with positive real parts, we acqu...

Here, we study the extension of p-trigonometric functions $\mathrm{sinp}$ and $\mathrm{cosp}$ family in complex domains and p-hyperbolic functions $\mathrm{sinhp}$ and the $\mathrm{coshp}$ family in hyperbolic complex domains. These functions satisfy analogous relations as their classical coun...

Determining the exact number of primes at large magnitudes is computationally intensive, making approximation methods (e.g., the logarithmic integral, prime number theorem, Riemann zeta function, Chebyshev’s estimates, etc.) particularly valuab...

This study deals with analytic functions with bounded turnings, defined in the disk $O_d=\left\{z:\left|z\right|<1\right\}$. These functions are subordinated by sigmoid function $\frac{2}{1+e^{-z}}$ and their class is denoted by ${\mathfrak{BT}}_{\mathfrak{Sg}}$. Sharp coefficient inequalities, including the third...

In this work, the Sieve of Eratosthenes procedure (in the following named Sieve procedure) is approached by a novel point of view, which is able to give a justification of the Prime Number Theorem (P.N.T.). Moreover, an extension of this procedure to...

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,...

Our main purpose in this paper is to study the anisotropic Moser–Trudinger-type inequalities with logarithmic weight ${\omega }_{\beta }(x)={[-ln{F}^o(x)|}^{(n-1)\beta }.$ This can be seen as a generation result of the isotropic Moser–Trudin...

The normalized analytic function ${\Phi }_N\left(z\right)=1+z-{\displaystyle \frac{z^3}{3}}$, which connects the open unit disk onto a bounded domain within the right half of a nephroid-shaped region, is associated with the bounded turning of functions denoted by $R_n$. It calculates the...

We present a method using contour integration to derive definite integrals and their associated infinite sums which can be expressed as a special function. We give a proof of the basic equation and some examples of the method. The advantage of using...