Related papers: More Jordan type inequalities

This paper investigates the analytic properties of the Liouville function's Dirichlet series that obtains from the function F(s)= zeta(2s)/zeta(s), where s is a complex variable and zeta(s) is the Riemann zeta function. The paper employs a…

A representation of the exceptional Lie algebras is presented. It reflects a simple unifying view and it is realized in terms of Zorn-type matrices. The role of the underlying Jordan pair and Jordan algebra content is crucial in the…

We introduce the zeta function of the prehomogenous vector space of binary cubic forms, twisted by the real analytic Eisenstein series. We prove the meromorphic continuation of this zeta function and identify its poles and their residues.…

Employing the Orlicz functions we extend the Buzano's inequality which is a refinement of the Cauchy-Schwarz inequality. Also using the Orlicz functions we obtain several numerical radius inequalities for a bounded linear operator as well…

In this paper, an elementary method to find the values of the Riemann Zeta function at even natural numbers, and to find values of a closely related series at odd natural numbers is presented. Another method, specifically for the evaluation…

We consider a Dirichlet series $\sum_{n=1}^{\infty}a_n^{-s}$, where $a_n$ satisfies a linear recurrence of arbitrary degree with integer coefficients. Under suitable hypotheses, we prove that it has a meromorphic continuation to the complex…

Unlike in complex linear operator theory, polynomials or, more generally, Laurent series in antilinear operators cannot be modelled with complex analysis. There exists a corresponding function space, though, surfacing in spectral mapping…

For an analytic function f(z)=z+\sum_{n=2}^\infty a_n z^n satisfying the inequality \sum_{n=2}^\infty n(n-1)|a_n|\leq \beta, sharp bound on $\beta$ is determined so that $f$ is either starlike or convex of order $\alpha$. Several other…

We use visible point vector identities to examine polylogarithms in the neighbourhood of the Riemann zeta function zeroes. New formulas limiting to the trivial zeroes and to the critical line on the zeta function are given. Similar results…

In this article we show a tecnique based on the Weierstrass product for the sine and cosine function and the bisection formula for the cotangent function that leads to a generalization of the classical Shafer-Fink inequality $ \frac{3…

In this paper the analytic characterization of generalized poles of operator valued generalized Nevanlinna functions (including the length of Jordan chains of the representing relation) is completed. In particular, given a Jordan chain of…

In this paper, we find some new sharp bounds for $\left(\sin x\right) /x$, which unify and refine Jordan, Adamovi\'{c}-Mitrinovi\'{c}and and Cusa's inequalities. As applications of main results, some new Shafer-Fink type inequalities for…

In this article, we study the local behaviour of the multiple polylogarithm functions at integer points, in the $s$-aspect. This is done by writing a Laurent type expansion at integer points, involving certain power series and rational…

We give a complete and elementary proofs of "Jordan's sums" and study Euler's types sums. In particular we give a formula for the sum of series with same weight, which is similar to this one of classical 2-Euler's sums.

The Zernike radial polynomials are a system of orthogonal polynomials over the unit interval with weight x. They are used as basis functions in optics to expand fields over the cross section of circular pupils. To calculate the roots of…

Let $\Re_n$ be the set of all rational functions of the type $r(z) = p(z)/w(z),$ where $p(z)$ is a polynomial of degree at most $n$ and $w(z) = \prod_{j=1}^{n}(z-a_j)$, $|a_j|>1$ for $1\leq j\leq n$. In this paper, we set up some results…

Jordan analytic curves which are invariant under rational functions are studied

Tangent numbers $T_{2n-1}$, which enumerate alternating permutations of odd length, play a prominent role in the Taylor series expansion of the tangent function $\tan(x)$. In this work, we adopt a combinatorial approach based on the…

In this note we prove Jensen-type inequality for certain non-convex functions. We apply our idea to prove some inequalities which were suggested at some high-level math olympiades.

We derive general formula for the fourth coefficient of the functions belonging to the Carath\'{e}odory class involving the parameters lying in the open unit disk. Further, we obtain sharp upper bounds of initial inverse coefficients for…