Coefficient inequalities for univalent starlike functions

Let A be the class of analytic functions in the unit disk $\mathbb{D}$ with the normalization f(0) = f′(0) ? 1 = 0. In this paper the authors discuss necessary and sufficient coefficient conditions for f ∈ A of the form $$\left( {\frac{z} {{f(z)}}} \right)^\mu = 1 + b_1 z + b_2 z^2 + \ldots$$ to be starlike in $\mathbb{D}$ and more generally, starlike of some order β, 0 ≤ β < 1. Here µ is a suitable complex number so that the right hand side expression is analytic in $\mathbb{D}$ and the power is chosen to be the principal power. A similar problem for the class of convex functions of order β is open.     

Coefficient inequalities for a subclass of starlike functions

A function f(z) = z ? ∑^∞_{n = 2}a_nzⁿ, a_n ? 0, analytic and univalent in the unit disk, is said to be in the family $\text{T}{}^1\text{(a, b)}$, a real and b ? 0, if $\text{¦(}\text{zf′}\text{f}\text{) ? a¦ ? b}$ for all z in the unit disk. A complete characterization is found for $\text{T}{}^1\text{(a, b)}$ when a ? 1. Also, sharp coefficient bounds are determined for certain subclasses of $\text{T}{}^1\text{(a, b)}$ when a < 1; however, examples are given to show that these bounds do not remain valid for the whole family.     

Some criteria for meromorphic multivalent starlike functions

The main purpose of the present paper is to derive some new criteria for meromorphic multivalent starlike functions.     

Some inequalities for characteristic functions

In a recent paper, Matysiak and Szablowski [V. Matysiak, P.J. Szablowski, Theory Probab. Appl. 45 (2001) 711-713] posed an interesting conjecture about a lower bound of real-valued characteristic functions. Under a suitable moment condition on distributions, we prove the conjecture to be true. The unified approach proposed here enables us to obtain new inequalities for characteristic functions. We also show by example that the improvement in the bounds is significant if more information about the distribution is available.     

Some inequalities for multivariate characteristic functions

We obtain some new lower and upper bounds for characteristic functions of multivariate distributions that can be useful in various applications. Supported by the Russian Foundation for Basic Research (grant Nos. 97-01-00273, 98-01-00621, and 98-01-00926) and by INTAS-RFBR (grant No. IR-97-0537). Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part II.     

Some sharp inequalities for n-monotone functions

Summary Some sharp inequalities involving n-monotone functions and their derivatives are obtained. In particular, the following generalization of the Favard-Berwald inequality is established here: \emph{If\/ is non-negative and -f is -monotone, then      

Some inequalities for gradients of harmonic functions

For a function u(x, y) harmonic in the upper half-plane y>0 and represented by the Poisson integral of a function v(t) ∈ L ₂ (−∞,∞), we prove that the inequality is true. A similar inequality is obtained for a function harmonic in a disk. Odessa Marine University, Odessa. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 8, pp. 1135–1136, August, 1997.     

Some integral inequalities for logarithmically convex functions

The main aim of the present note is to establish new Hadamard like integral inequalities involving log-convex function. We also prove some Hadamard-type inequalities, and applications to the special means are given.     

Some argument inequalities for certain analytic functions

For analytic functions f(z) in the open unit disk U and convex functions g(z) in U, Nunokawa et al. [NUNOKAWA, M.—OWA, S.—NISHIWAKI, J.—KUROKI, K.—HAYAMI, T: Differential subordination and argumental property, Comput. Math. Appl. 56 (2008), 2733–2736] have proved one theorem which is a generalization of the result [POMMERENKE, CH.: On close-toconvex analytic functions, Trans. Amer. Math. Soc. 114 (1965), 176–186]. The object of the present paper is to generalize the theorem due to Nunokawa et al..     

Some inequalities for characteristic functions. II

Recently we established Matysiak and Szablowski's conjecture [V. Matysiak, P.J. Szablowski, Theory Probab. Appl. 45 (2001) 711-713] about a lower bound of real-valued characteristic functions. In this paper, applying an alternative approach we are able to give explicitly the ranges of argument for which the obtained inequalities hold true for general characteristic functions.     

Some divided difference inequalities for n-convex functions

Using results from the theory of B-splines, various inequalities involving the nth order divided differences of a function f with convex nth derivative are proved; notably, $\text{f}{}^{\mathrm{(n)}}\text{(z)}\text{n! ? [x}{}_0\text{,…, x}{}_{\mathrm{n}}\text{]f}\text{?}\text{∑}{}_{\mathrm{i\; =\; 0}}{}^{\mathrm{n}}\text{(f}{}^{\mathrm{(n)}}\text{(x}{}_{\mathrm{i}}\text{)}\text{(n + 1)!)}$, where z is the center of mass $\text{(}\text{1}\text{(n + 1))}\text{∑}{}_{\mathrm{i\; =\; 0}}{}^{\mathrm{n}}\text{x}{}_{\mathrm{i}}$.     

Some quantum estimates of Hermite-Hadamard inequalities for convex functions

In this study, based on a new quantum integral identity, we establish some quantum estimates of Hermite-Hadamard type inequalities for convex functions. These results generalize and improve some known results given in literatures.