Two-sided Inequalities Research Articles

Coefficient and Fekete-Szegö problem estimates for certain subclass of analytic and bi-univalent functions

In this paper, we obtain the Fekete-Szeg? problem for the k-th (k ? 1) root transform of the analytic and normalized functions f satisfying the condition 1+ ? ??/2sin? < Re{z f'(z)/f(z)) < 1+ ?/2sin? (|z| < 1), where ? ? [?/2,?). Afterwards, by the above two-sided inequality we introduce a certain subclass of analytic and bi-univalent functions in the disk |z| < 1 and obtain upper bounds for the first few coefficients and Fekete-Szeg? problem for functions f belonging to this class.

An exact decomposition algorithm for a chance-constrained new product risk model

We use the generalized two-sided Chebyshev inequality to reformulate a certain nonlinear, chance-constrained new product risk model. The problem has a linear cost objective and a constraint set featuring a probabilistic lower bound on an event which depends on a collection of mutually-independent, uniform random parameters. Our reformulation permits a reduction of the problem to a sequence of second-order cone programs. We, therefore, identify a new family of non-convex programs whose members are amenable to convex programming solution techniques.

The multiple Cantelli inequalities

The Cantelli inequality or the one-sided Chebyshev inequality is extended to the problem of the probability of multiple inequalities for events with more than one variable. The corresponding two-sided Cantelli inequality is extended in a similar manner. The results for the linear combination of several variables are also given as their special cases.

Relative deviation learning bounds and generalization with unbounded loss functions

We present an extensive analysis of relative deviation bounds, including detailed proofs of two-sided inequalities and their implications. We also give detailed proofs of two-sided generalization bounds that hold in the general case of unbounded loss functions, under the assumption that a moment of the loss is bounded. We then illustrate how to apply these results in a sample application: the analysis of importance weighting.

On certain new means generated by generalized trigonometric functions

In this paper, authors generalize logarithmic mean $L$, Neuman-Sándor $M$, two Seiffert means $P$ and $T$ as an application of generalized trigonometric and hyperbolic functions. Moreover, several two-sided inequalities involving these generalized means are established.

Counting Unlabeled Bipartite Graphs Using Polya's Theorem

This paper solves a problem that was stated by M. A. Harrison in 1973. The problem has remained open since then, and it is concerned with counting equivalence classes of $n\times r$ binary matrices under row and column permutations. Let $I$ and $O$ denote two sets of vertices, where $I\cap O =\emptyset$, $|I| = n$, $|O| = r$, and $B_u(n,r)$ denote the set of unlabeled graphs whose edges connect vertices in $I$ and $O$. Harrison established that the number of equivalence classes of $n\times r$ binary matrices is equal to the number of unlabeled graphs in $B_u(n,r).$ He also computed the number of such matrices (hence such graphs) for small values of $n$ and $r$ without providing an asymptotic formula for $|B_u(n,r)|.$ Here, such an asymptotic formula is provided by proving the following two-sided inequality using Polya's Counting Theorem. \begin{equation} \displaystyle \frac{\binom{r+2^{n}-1}{r}}{n!} \le |B_u(n,r)| \le 2\frac{\binom{r+2^{n}-1}{r}}{n!}, n< r. \end{equation}

Estimates of the mean size of the subset image under composition of random mappings

AbstractLet XNbe a set ofNelements andF1,F2,… be a sequence of random independent equiprobable mappings XN→N. For a subsetS0⊂ XN, |S0|=m, we consider a sequence of its imagesSt=Ft(…F2(F1(S0))…),t=1,2… An approach to the exact recurrent computation of distribution of |St| is described. Two-sided inequalities forM{|St|||S0|=m} such that the difference between the upper and lower bounds iso(m)form,t,N→ ∞,mt=o(N) are derived. The results are of interest for the analysis of time-memory tradeoff algorithms.

New integral representations for the Fox–Wright functions and its applications

Our aim in this paper is to derive several new integral representations for the Fox–Wright functions. In particular, we give new Laplace and Stieltjes transforms for this special function under some restrictions on parameters. From the positivity conditions on the weight in these representations, we found sufficient conditions to be imposed on the parameters of the Fox–Wright functions which allow us to conclude that it is completely monotonic. As applications, we derive a class of functions that are related to the Fox H-functions and are positive definite. Moreover, we extended the Luke's inequalities and we establish new Turán type inequalities for the Fox–Wright function. Finally, by appealing to each of the Luke's inequalities, two sets of two-sided bounding inequalities for the generalized Mathieu's type series are proved.

Bounds for modified Struve functions of the first kind and their ratios

We obtain a simple two-sided inequality for the ratio Lν(x)/Lν−1(x) in terms of the ratio Iν(x)/Iν−1(x), where Lν(x) is the modified Struve function of the first kind and Iν(x) is the modified Bessel function of the first kind. This result allows one to use the extensive literature on bounds for Iν(x)/Iν−1(x) to immediately deduce bounds for Lν(x)/Lν−1(x). We note some consequences and obtain further bounds for Lν(x)/Lν−1(x) by adapting techniques used to bound the ratio Iν(x)/Iν−1(x). We apply these results to obtain new bounds for the condition numbers xLν′(x)/Lν(x), the ratio Lν(x)/Lν(y) and the modified Struve function Lν(x) itself. Amongst other results, we obtain two-sided inequalities for xLν′(x)/Lν(x) and Lν(x)/Lν(y) that are given in terms of xIν′(x)/Iν(x) and Iν(x)/Iν(y), respectively, which again allows one to exploit the substantial literature on bounds for these quantities. The results obtained in this paper complement and improve existing bounds in the literature.

Barvinok’s naive algorithm in Distance Geometry

In 1997, A. Barvinok gave a probabilistic algorithm to derive a near-feasible solution of a quadratically (equation) constrained problem from its semidefinite relaxation. We generalize this algorithm to handle matrix variables instead of vectors, and to handle two-sided inequalities instead of equations. We derive a heuristic for the distance geometry problem, and showcase its computational performance on a set of instances related to protein conformation.

On a New Subclass of p-Valent Close-to-Convex Mappings Defined by Two-Sided Inequality

A new subclass Kp(α1,α2,β) of p-valent close-to-convex mappings defined by two-sided inequality is introduced. Some sufficient conditions for functions to be in Kp(α1,α2,β) are given.

Two-sided inequalities for the Struve and Lommel functions

Mathematical inequalities and other results involving such widely- and extensively-studied special functions of mathematical physics and applied mathematics as (for example) the Bessel, Struve and Lommel functions as well as the associated hypergeometric functions are potentially useful in many seemingly diverse areas of applications, especially in situations in which these functions are involved in solutions of mathematical, physical and engineering problems which can be modeled by ordinary and partial differential equations. With this objective in view, our present investigation is motivated by some open problems involving inequalities for a number of particular forms of the hypergeometric function 1F2(a; b, c; z). Here, in this paper, we apply a novel approach to such problems and obtain presumably new two-sided inequalities for the Struve function, the associated Struve function and the modified Struve function by first investigating inequalities for the general hypergeometric function 1F2(a; b, c; z). We also briefly discuss the analogous new inequalities for the Lommel function under some conditions and constraints. Finally, as special cases of our main results, we deduce several inequalities for the modified Lommel function and the normalized Lommel function.

Two-Sided Inequality Involving the q-Gamma Function

An inequality involving the q-gamma and q-trigamma functions is derived and proved for all real number $$q>0$$ . This inequality provides new bounds for the q-gamma function in terms of the q-trigamma function.

Inequalities and asymptotics for some moment integrals

For alpha>beta-1>0, we obtain two-sided inequalities for the moment integral I(alpha,beta)=int_{mathbb{R}}|x|^{-beta}|sin x|^{alpha},dx. These are then used to give the exact asymptotic behavior of the integral as alpharightarrowinfty. The case I(alpha,alpha) corresponds to the asymptotics of Ball’s inequality, and I(alpha,[alpha]-1) corresponds to a kind of novel “oscillatory” behavior.

Some new two-sided inequalities concerning the Fourier transform

Some new two-sided inequalities concerning the Fourier transform

Estimates for Wallis’ ratio and related functions

We present improvements of approximation formula for Wallis ratio related to a class of inequalities stated in [D.-J. Zhao, On a two-sided inequality involving Wallis’s formula, Math. Practice Theory, 34 (2004), 166-168], [Y. Zhao and Q. Wu, Wallis inequality with a parameter, J. Inequal. Pure Appl. Math., 7(2) (2006), Art. 56] and [C. Mortici, Completely monotone functions and the Wallis ratio, Applied Mathematics Letters, 25 (2012), 717-722]. Some sharp inequalities are obtained as a result of monotonicity of some functions involving gamma function.

Some new two-sided inequalities concerning the Fourier transform

Acknowledgement. The authors thank the careful referee for some generous suggestions, which have improvedthe final version of this paper. The authors thank Lule˚a University of Technology for financial support for the research visit of the first authorin December 2015. The research of the first author was also supported by the MESRK grants 5540/GF4 and the research of the second author was supportedby the MESRK grants4080/GF4. The publication was supported by the Ministry of Education and Science of the Russian Federation (the Agreement number02.a03.21.0008).

Two-sided inequalities with nonmonotone sublinear operators

The theorems on solutions and their two-sided estimates for one class of nonlinear operator equations $x=Fx$ with nonmonotone operators.

Representations and Inequalities for Generalized Hypergeometric Functions

An integral representation for the generalized hypergeometric function unifying known representations via generalized Stieltjes, Laplace, and cosine Fourier transforms is found. Using positivity conditions for the weight in this representation, various new facts regarding generalized hypergeometric functions, including complete monotonicity, log-convexity in upper parameters, monotonicity of ratios, and new proofs of Luke’s bounds are established. In addition, two-sided inequalities for the Bessel type hypergeometric functions are derived with the use of their series representations. Bibliography: 22 titles.

A sharp two-sided inequality for bounding the Wallis ratio

In this article we establish a sharp two-sided inequality for bounding the Wallis ratio. Some best constants for the estimation of the Wallis ratio are obtained. An asymptotic formula for the Wallis ratio is also presented.