For a convex domain $K$ in the complex plane, the well-known general Bernstein-Markov inequality holds asserting that a polynomial $p$ of degree $n$ must have $||p'|| < c(K) n^2 ||p||$. On the other hand for polynomials in general, $||p'||$ can be arbitrarily small as compared to $||p||$.
The situation changes when we assume that the polynomials in question have all their zeroes in the convex body $K$. This was first investigated by Turán, who showed the lower bounds $||p'|| \ge (n/2) ||p||$ for the unit disk $D$ and $||p'|| > c \sqrt{n} ||p||$ for the unit interval $I:=[-1,1]$. Although partial results provided general lower estimates of lower order, as well as certain classes of domains with lower bounds of order $n$, it was not clear what order of magnitude the general convex domains may admit here.
Here we show that for all compact and convex domains $K$ with nonempty interior and polynomials $p$ with all their zeroes in $K$ $||p'|| > c(K) n ||p||$ holds true, while $||p'|| < C(K) n ||p||$ occurs for any $K$. Actually, we determine $c(K)$ and $C(K)$ within a factor of absolute numerical constant.
We prove turán type inequalities for Dunkl kernel. We provide a $q$-integral representation for the $q$-Dunkl kernel. Using a $q$-version of Schwartz inequality, we get a turán type inequalities for $q$-Dunkl kernel.
In this paper our aim is to deduce some sharp Turán type inequalities for the remainder $q-$exponential functions. Our results are shown to be a generalization of results which were obtained by Alzer \cite{al}.
In this paper we deduce some tight Turán type inequalities for Tricomi confluent hypergeometric functions of the second kind, which in some cases improve the existing results in the literature. We also give alternative proofs for some already established Turán type inequalities. Moreover, by using these Turán type inequalities, we deduce some new inequalities for Tricomi confluent hypergeometric functions of the second kind. The key tool in the proof of the Turán type inequalities is an integral representation for a quotient of Tricomi confluent hypergeometric functions, which arises in the st…
We establish some new Turán's type inequalities for orthogonal polynomials defined by a three-term recurrence with monotonic coefficients. As a corollary we deduce asymptotic bounds on the extreme zeros of orthogonal polynomials with polynomially growing coefficients of the three-term recurrence.
In this paper some Turán type inequalities for the general Bessel function, monotonicity and bounds for its logarithmic derivative are derived. Moreover we find the series representation and the relative extrema of the Turánian of general Bessel functions. The key tools in the proofs are the recurrence relations together with some asymptotic relations for Bessel functions.
In this paper we study the inverse of the eigenfunction $\sin_p$ of the one-dimensional $p$-Laplace operator and its dependence on the parameter $p$, and we present a Turán type inequality for this function. Similar inequalities are given also for other generalized inverse trigonometric and hyperbolic functions. In particular, we deduce a Turán type inequality for a series considered by Ramanujan, involving the digamma function.
In this paper some Turán type inequalities for classical and generalized Mittag-Leffler functions are considered. The method is based on proving monotonicity for special ratio of sections for series of Mittag-Leffler functions. Some applications are considered to Lazarević and Wilker-type inequalities for classical and generalized Mittag-Leffler functions. At the end some open problems are formulated.
Turán, Mitrinović-Adamović and Wilker type inequalities are deduced for regular Coulomb wave functions. The proofs are based on a Mittag-Leffler expansion for the regular Coulomb wave function, which may be of independent interest. Moreover, some complete monotonicity results concerning the Coulomb zeta functions and some interlacing properties of the zeros of Coulomb wave functions are given.
Complete monotonicity, Laguerre and Turán type inequalities are established for the so-called Krätzel function $Z_ρ^ν,$ defined by $$Z_ρ^ν(u)=\int_0^{\infty}t^{ν-1}e^{-t^ρ-\frac{u}{t}}\dt,$$ where $u>0$ and $ρ,ν\in\mathbb{R}.$ Moreover, we prove the complete monotonicity of a determinant function of which entries involve the Krätzel function.
In this note our aim is to point out that certain inequalities for modified Bessel functions of the first and second kind, deduced recently by Laforgia and Natalini, are in fact equivalent to the corresponding Turán type inequalities for these functions. Moreover, we present some new Turán type inequalities for the aforementioned functions and we show that their product is decreasing as a function of the order, which has application in the study of stability of radially symmetric solutions in a generalized FitzHugh-Nagumo equation in two spatial dimensions. At the end of this note a conjecture…
In this paper we prove monotonicity of some ratios of $q$--Kummer confluent hypergeometric and $q$--hypergeometric functions. The results are also closely connected with Turán type inequalities. In order to obtain main results we apply methods developed earlier by the authors for the case of classical Kummer and Gauss hypergeometric functions.
In this paper certain Turán type inequalities for some Lommel functions of the first kind are deduced. The key tools in our proofs are the infinite product representation for these Lommel functions of the first kind, a classical result of G. Pólya on the zeros of some particular entire functions, and the connection of these Lommel functions with the so-called Laguerre-Pólya class of entire functions. Moreover, it is shown that in some cases J. Steinig's results on the sign of Lommel functions of the first kind combined with the so-called monotone form of l'Hospital's rule can be used in the pr…
The paper studies logarithmic convexity and concavity of the generalized hypergeometric function with respect to simultaneous shift of several parameters. We use integral representations and properties of Meijer's $G$ function to prove log-convexity. When all parameters are shifted we use series manipulations to examine the power series coefficients of the generalized Turánian formed by the generalized hypergeometric function. In cases when all zeros of the generalized hypergeometric function are real, we further explore the consequences of the extended Laguerre inequalities and formulate a co…
This paper studies the log-convexity of the extended beta functions. As a consequence, Turán-type inequalities are established.The monotonicity, log-convexity, log-concavity of extended hypergeometric functions are deduced by using the inequalities on extended beta functions,. The particular cases of those results also gives the Turán-type inequalities for extended confluent and extended Gaussian hypergeometric functions. Some reverse of Turán-type inequalities also derived.
Some Turán type inequalities for Struve functions of the first kind are deduced by using various methods developed in the case of Bessel functions of the first and second kind. New formulas, like Mittag-Leffler expansion, infinite product representation for Struve functions of the first kind, are obtained, which may be of independent interest. Moreover, some complete monotonicity results and functional inequalities are deduced for Struve functions of the second kind. These results complement naturally the known results for a particular case of Lommel functions of the first kind, and for modifi…
Some sharp two-sided Turán type inequalities for parabolic cylinder functions and Tricomi confluent hypergeometric functions are deduced. The proofs are based on integral representations for quotients of parabolic cylinder functions and Tricomi confluent hypergeometric functions, which arise in the study of the infinite divisibility of the Fisher-Snedecor F distribution. Moroever, some complete monotonicity results are given concerning Turán determinants of Tricomi confluent hypergeometric functions. These complement and improve some of the results of Ismail and Laforgia [23].
In 1939 P. Turán started to derive lower estimations on the norm of the derivatives of polynomials of (maximum) norm 1 on $I:=[-1,1]$ (interval) and $D:=\{z\in\mathbb{C}~:~|z|\le 1\}$ (disk), under the normalization condition that the zeroes of the polynomial in question all lie in $I$ or $D$, respectively. For the maximum norm he found that with $n:=\mathop{\mathrm{deg}} p$ tending to infinity, the precise growth order of the minimal possible derivative norm is $\sqrt{n}$ for $I$ and $n$ for $D$. J. Erőd continued the work of Turán considering other domains. Finally, a decade ago the growth o…
In this paper first we survey the Turán type inequalities and related problems for the Bessel functions of the first kind. Then we extend the known higher order Turán type inequalities for Bessel functions of the first kind to real parameters and we deduce new closed integral representation formulae for the second kind Neumann type series of Bessel functions of the first kind occurring in the study of Turán determinants of Bessel functions of the first kind. At the end of the paper we prove a Turán type inequality for the Bessel functions of the second kind.
In this present paper, we establish the log-convexity and Turán type inequalities of extended $(p,q)$-beta functions. Also, we present the log-convexity, the monotonicity and Turán type inequalities for extended $(p,q)$-confluent hypergeometric function by using the inequalities of extended $(p,q)$-beta functions.
Let $g$ and $h$ be real-valued arithmetic functions, positive and normalized. Specific choices within the following general scheme of recursively defined polynomials \begin{equation*} P_n^{g,h}(x):= \frac{x}{h(n)} \sum_{k=1}^{n} g(k) \, P_{n-k}^{g,h}(x), \end{equation*} with initial value $P_{0}^{g,h}(x)=1$ encode information about several classical, widely studied polynomials. This includes Chebyshev polynomials of the second kind, associated Laguerre polynomials, and the Nekrasov--Okounkov polynomials. In this paper we prove that for $g(n)=n$ and fixed $h$ we obtain orthogonal polynomial seq…
For rational functions, we use simple but elegant techniques to strengthen generalizations of certain results which extend some widely known polynomial inequalities of Erdös-Lax and Turán to rational functions R. In return these reinforced results, in the limiting case, lead to the corresponding refinements of the said polynomial inequalities. As an illustration and as an application of our results, we obtain some new improvements of the Erdós-Lax and Turán type inequalities for polynomials. These improved results take into account the size of the constant term and the leading coefficient of t…
In this paper, our aim is to show some mean value inequalities for the Fox-Wright functions, such as Turán--type inequalities, Lazarević and Wilker--type inequalities. As applications we derive some new type inequalities for hypergeometric functions and the four--parametric Mittag--Leffler functions. Furthermore, we prove monotonicity of ratios for sections of series of Fox-Wright functions, the results is also closely connected with Turán--type inequalities. Moreover, some other type inequalities are also presented. At the end of the paper, some problems stated, which may be of interest for f…
We show how Turan's inequality $P_n(x)^2-P_{n-1}(x)P_{n+1}(x)\geq 0$ for Legendre polynomials and related inequalities can be proven by means of a computer procedure. The use of this procedure simplifies the daily work with inequalities. For instance, we have found the stronger inequality $|x|P_n(x)^2-P_{n-1}(x)P_{n+1}(x)\geq 0$, $-1\leq x\leq 1$, effortlessly with the aid of our method.
This article studies the monotonicity, log-convexity of the modified Lommel functions by using its power series and infinite product representation. Same properties for the ratio of the modified Lommel functions with the Lommel function, $\sinh$ and $\cosh$ are also discussed. As a consequence, some Turán type and reverse Turán type inequalities are given. A Rayleigh type function for the Lommel functions are derived and as an application, we obtain the Redheffer-type inequality.
Inspired by the work of C. Mortici [1] and A. Laforgia et. al [2] we have established some new Turán-type inequalities for k-polygamma function and p-k-polygamma function.
Turán type oscillation inequalities in $L^q$ norm on the boundary of convex polygonal domains
Institute of Natural Sciences and Mathematics（自然科学与数学研究所）；Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences（俄罗斯科学院乌拉尔分支数学力学研究所）；HUN-REN A. Rényi Institute of Mathematics（匈牙利研究与教育网络阿尔弗雷德·雷尼数学研究所）
In 1935, P. Turán proved that $$ S_{n,a}(x)= \sum_{j=1}^n{n+a-j\choose n-j} \sin(jx)>0 \quad{(n,a\in\mathbf{N}; 0<x<π).} $$ We present various related inequalities. Among others, we show that the refinements $$ S_{2n-1,a}(x)\geq \sin(x) \quad\mbox{and} \quad{S_{2n,a}(x)\geq 2\sin(x)(1+\cos(x))} $$ are valid for all integers $n\geq 1$ and real numbers $a\geq 1$ and $x\in(0,π)$. Moreover, we apply our theorems on sine sums to obtain inequalities for the Chebyshev polynomials of the second kind.
For a convex domain K in the complex plane C, the well-known general Markov inequality asserting that a polynomial p of degree n ||p'|| < c(K) n^2 ||p|| holds. On the other hand for polynomials in general, ||p'|| can be arbitrarily small as compared to ||p||. The situation changes when we assume that the polynomials have all their zeroes in the convex body K. This problem of lower bound for Markov factors was first investigated by Turán in 1939. Turán showed ||p'|| \ge n/2 ||p|| for the unit disk D and ||p'|| > c \sqrt{n} ||p|| for the unit interval I:=[-1,1]. Soon after that, J. Er\H od publi…
Let ${\cal P}_n^c$ denote the set of all algebraic polynomials of degree at most $n$ with complex coefficients. Let $$D^+ := \{z \in \mathbb{C}: |z| \leq 1, \, \, \Im(z) \geq 0\}$$ be the closed upper half-disk of the complex plane. For integers $0 \leq k \leq n$ let ${\mathcal F}_{n,k}^c$ be the set of all polynomials $P \in {\mathcal P}_n^c$ having at least $n-k$ zeros in $D^+$. Let $$\|f\|_A := \sup_{z \in A}{|f(z)|}$$ for complex-valued functions defined on $A \subset {\Bbb C}$. We prove that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that $$c_1 \left(\frac{n}{k+1}\right)^{1…
It is demonstrated that multiplier methods naturally yield better constants in strong converse inequalities for the Bernstein-Durrmeyer operator. The absolute constants obtained in some of the inequalities are independent of the weight and the dimension. The estimates are stated in terms of the $K$-functional that is naturally associated to the operator.
Our aim in this paper is to show some new inequalities for Mathieu's type series and Riemann zeta function. In particular, some Turán type inequalities, some monotonicity and log-convexity results for these special functions are given. New Laplace type integral representations for Mathie type series and Riemann zeta function are also presented.
We deduce some new functional inequalities, like Turán type inequalities, Redheffer type inequalities, and a Mittag-Leffler expansion for a special combination of modified Bessel functions of the first kind, called modified Dini functions. Moreover, we show the complete monotonicity of a quotient of modified Dini functions by introducing a new continuous infinitely divisible probability distribution. The key tool in our proofs is a recently developed infinite product representation for a special combination of Bessel functions of the first, which was very useful in determining the radius of co…
We prove two inequalities for the Mittag-Leffler function, namely that the function $\log E_α(x^α)$ is sub-additive for $0<α<1,$ and super-additive for $α>1.$ These assertions follow from two new binomial inequalities, one of which is a converse to the neo-classical inequality. The proofs use a generalization of the binomial theorem due to Hara and Hino (Bull. London Math. Soc. 2010). For $0<α<2,$ we also show that $E_α(x^α)$ is log-concave resp. log-convex, using analytic as well as probabilistic arguments.
Approximation on the spherical cap is different from that on the sphere which requires us to construct new operators. This paper discusses the approximation on the spherical cap. That is, so called Jackson-type operator $\{J_{k,s}^m\}_{k=1}^{\infty}$ is constructed to approximate the function defined on the spherical cap $D(x_0,γ)$. We thus establish the direct and inverse inequalities and obtain saturation theorems for $\{J_{k,s}^m\}_{k=1}^{\infty}$ on the cap $D(x_0,γ)$. Using methods of $K$-functional and multiplier, we obtain the inequality \begin{eqnarray*} C_1\:\| J_{k,s}^m(f)-f\|_{D,p}\…
In this paper we point out a converse result of the celebrated Jensen inequality for differentiable convex mappings of several variables and apply it to counterpart well-known analytic inequalities. Applications to Shannon's and Renyi's entropy mappings are also given.
Turán type inequalities for modified Bessel functions of the first kind are used to deduce some sharp lower and upper bounds for the asymptotic order parameter of the stochastic Kuramoto model. Moreover, approximation from the Lagrange inversion theorem and a rational approximation are given for the asymptotic order parameter.
It is shown that $\sum_{j=-m}^m (-1)^j \frac{f(x-j)(f(x+j)}{(m-j)! (m+j)!} \ge 0,$ $m=0,1,...,$ where $f(x)$ is either a real polynomial with only real zeros or an allied entire function of a special type, provided the distance between two consecutive zeros of $f(x)$ is at least $\sqrt{4-\frac{6}{m+2}}.$ These inequalities are a surprisingly similar discrete analogue of higher degree generalizations of the Laguerre and Turan inequalities. Being applied to the classical discrete orthogonal polynomials, they yield sharp, explicit bounds uniform in all parameters involved, on the polynomials and …
We obtain extremal majorants and minorants of exponential type for a class of even functions on $\R$ which includes $\log |x|$ and $|x|^α$, where $-1 < α< 1$. We also give periodic versions of these results in which the majorants and minorants are trigonometric polynomials of bounded degree. As applications we obtain optimal estimates for certain Hermitian forms, which include discrete analogues of the one dimensional Hardy-Littlewood-Sobolev inequalities. A further application provides an Erdös-Turán-type inequality that estimates the sup norm of algebraic polynomials on the unit disc in term…
We consider certain inequalities among the Apollonian metric, the Apollonian inner metric, the $j$ metric and the quasihyperbolic metric. We verify that whether these inequalities can occur in simply connected planar domains and in proper subdomains of $\mathbb{R}^n~(n\ge 2)$. We have seen from our verification that most of the cases cannot occur. This means that there are many restrictions on domains in which these inequalities can occur. We also consider two metrics $j$ and $d$, and investigate whether a plane domain $D\varsubsetneq\mathbb{C}$, for which there exists a constant $c>0$ with $j…
This paper introduces and studies a generalization of the $\mathtt{k}$-Bessel function of order $ν$ given by \[\mathtt{W}^{\mathtt{k}}_{ν, c}(x):= \sum_{r=0}^\infty \frac{(-c)^r}{Γ_{\mathtt{k}}\left( r \mathtt{k} +ν+\mathtt{k}\right) r!} \left(\frac{x}{2}\right)^{2r+\fracν{\mathtt{k}}}. \] Representation formulae are derived for $\mathtt{W}^{\mathtt{k}}_{ν, c}.$ Further the function $\mathtt{W}^{\mathtt{k}}_{ν, c}$ is shown to be a solution of a second order differential equation. Monotonicity and log-convexity properties for the generalized $\mathtt{k}$-Bessel function $\mathtt{W}^{\mathtt{k}…
