http://www.hindawi.comThe latest articles from Hindawi Publishing Corporation© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/697407A measure of noncompactness (essential norm) for maximal functions and potential operators defined on homogeneous groups is estimated in terms of weights. Similar problem for partial sums of the Fourier series is studied. In some cases, we conclude that there is no weight pair for which these operators acting between two weighted Lebesgue spaces are compact.Muhammad Asif and Alexander Meskhi© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/420989We study the strong convergence of an iterative method for inverse strongly accretive operators in the framework of Banach spaces. Our results improve and extend the corresponding results announced by many others.Yan Hao© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/451815We consider some classes of 2π-periodic convolution functions B˜p, and K˜p, which include the classical Sobolev class as a special case. With the help of the spectra of nonlinear integral equations, we determine the exact values of Bernstein n-width of the classes B˜p, K˜p in the space Lp for 1<p<∞.Feng Guo© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/389410We give some further consideration about logarithmic convexity for differences of power sums inequality as well as related mean value theorems. Also we define quasiarithmetic sum and give some related results.J. Pečarić and Atiq Ur Rehman© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/598319For a sequence {Xn,n≥1} of dependent square integrable random variables and a sequence {bn,n≥1} of positive numbers, we establish a maximal inequality for weighted sums of dependent random variables. Applying this inequality, we obtain the almost sure convergence of ∑i=1nXi/bi and ∑i=1nXi/bn.Soo Hak Sung© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/894529We consider the classes of periodic functions with formal self-adjoint linear differential operators Wp(ℒr), which include the classical Sobolev class as its special case. With the help of the spectral of linear differential equations, we find the exact values of Bernstein n-width of the classes Wp(ℒr) in the Lp for 1<p<∞.Feng Guo© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/656329We define two functions which determine the properties and the representation of the mean value of a function on a ball and on its associated sphere. Using these two functions, we obtain Pizzetti's formula in Rn as well as a similar formula for the mean value of a function on the ball associated to the sphere. We also give the expressions of the remainders in these two formulas, using the surface integral on a sphere.Antonela Toma© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/717614Suppose that f(x) is strictly increasing, strictly concave, and twice continuously differentiable on a nonempty interval I⊆ℝ, and f′(x) is strictly convex on I. Suppose that xk∈[a,b]⊆I, where 0<a<b, and pk≥0 for k=1,⋯,n, and suppose that ∑k=1npk=1. Let x̄=∑k=1npkxk, and σ2=∑k=1npk(xk−x̄)2. We show ∑k=1npkf(xk)≤f(x̄−θ1σ2), ∑k=1npkf(xk)≥f(x̄−θ2σ2), for suitably chosen θ1 and θ2. These results can be viewed as a refinement of the Jensen's inequality for the class of functions specified above. Or they can be viewed as a generalization of a refined arithmetic mean-geometric mean inequality introduced by Cartwright and Field in 1978. The strength of the above result is in bringing the variations of the xk's into consideration, through σ2.Ye Xia© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/715961The Brézis-Wainger inequality on a compact Riemannian manifold without boundary is shown. For this purpose, the Moser-Trudinger inequality and the Sobolev embedding theorem are applied.Przemysław Górka© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/127645In this paper we have introduced an integral general operator. For this general operator which is a generalization of more known integral operators we have demonstrated some univalence properties.Georgia Irina Oros, Gheorghe Oros, and Daniel Breaz© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/917392By introducing an integral operator, a norm with a weight function, and two pairs of conjugate exponents, we find the conditions for the Hilbert-type integral inequalities with a homogeneous kernel of −λ-degree. We also prove that the constant factors in the inequalities are all the best possible. As the particular situations, some new inequalities with a homogeneous kernel and their other two forms are given. We extend some previous results.Wuyi Zhong© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/690932Let B be the unit ball in ℂn with respect to the Euclidean norm. In this paper, we obtain a sufficient condition for a normalized quasiregular mapping f∈H(B) to be extended to a quasiconformal homeomorphism of ℝ2n onto itself. In the last section we consider the asymptotical case of this result and we obtain certain applications.Paula Curt and Gabriela Kohr© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/325845A multidimensional recurring mean inequality is shown. Furthermore, we prove some new inequalities, which can be considered to be the extensions of those established inequalities, including, for example, the Polya-Szegö and Kantorovich inequalities .Mingjin Wang© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/249438The main purpose of the present paper is to establish some new Grüss integral inequalities in n independent variables. Our results in special cases yield some of the recent results on Pachpatte's, Mitrinović's, and Ostrowski's inequalities, and provide new estimates on such types of inequalities.Chang-Jian Zhao and Wing-Sum Cheung© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/702715In this paper we considered an general integral operator and three classes of univalent functions for which the second order derivative is equal to zero. By imposing supplimentary conditions for these functions we proved some univalent conditions for the considered general operator. Also some interesting particullar results are presented.Daniel Breaz and H. Özlem Güney© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/637397It is shown that a weighted Hilbert inequality for double series can be established by introducing a proper weight function. Thus, a quite sharp result of the classical Hilbert inequality for double series is obtained. And a similar result for the Hilbert integral inequality is also proved. Some applications are considered.He Leping, Gao Xuemei, and Gao Mingzhe© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/280183The aim of this paper is to give the radius of starlikeness of the certain classes of p-valent functions defined by multiplier transformations. The results are obtained by using techniques of Robertson (1953,1963) which was used by Bernardi (1970), Libera (1971), Livingstone (1966), and Goel (1972).Mugur Acu, Yaşar Polatog̃lu, and Emel Yavuz© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/278505We first examine the generalized Hyers-Ulam stability of functional inequality associated with module Jordan left derivation (resp., module Jordan derivation). Secondly, we study the functional inequality with linear Jordan left derivation (resp., linear Jordan derivation) mapping into the Jacobson radical.Hark-Mahn Kim, Sheon-Young Kang, and Ick-Soon Chang© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/374756The authors investigate various subordination results for some subclasses of analytic functions in the unit disc. We obtain some sufficient conditions for multivalent close-to-starlikeness.Öznur Özkan Kılıç© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/390435The present article investigates new classes of functions involving generalized Noor integral operator. Some properties of these functions are studied including characterization and distortion theorems. Moreover, we illustrate sufficient conditions for subordination and superordination for analytic functions.Rabha W. Ibrahim and Maslina Darus© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/518646This paper improves Pachpatte's results on linear integral inequalities with two variables, and gives an estimation for a general form of nonlinear integral inequality with two variables. This paper does not require monotonicity of known functions. The result of this paper can be applied to discuss on boundedness and uniqueness for a integrodifferential equation.Wu-Sheng Wang and Cai-Xia Shen© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/571417The main purpose of the present paper is to establish some new sharp integral inequalities involving higher-order partial derivatives. Our results in special cases yield some of the recent results on Agarwal, Wirtinger, Poincaré, Pachpatte, Smith, and Stredulinsky's inequalities and provide some new estimates on such types of inequalities.C.-J Zhao and W.-S Cheung© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/576876The theory and applications of dynamic derivatives on time scales have recently received considerable attention. The primary purpose of this paper is to give basic properties of diamond-α derivatives which are a linear combination of delta and nabla dynamic derivatives on time scales. We prove a generalized version of Jensen's inequality on time scales via the diamond-α integral and present some corollaries, including Hölder's and Minkowski's diamond-α integral inequalities.Moulay Rchid Sidi Ammi, Rui A. C. Ferreira, and Delfim F. M. Torres© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/210626We study the following generalized additive functional inequality ‖af(x)+bf(y)+cf(z)‖≤‖f(αx+βy+γz)‖, associated with linear mappings in Banach spaces. Moreover, we prove the Hyers-Ulam-Rassias stability of the above generalized additive functional inequality, associated with linear mappings in Banach spaces.Jung Rye Lee, Choonkil Park, and Dong Yun Shin© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/904852We investigate the following functional inequality: ‖f(x)+f(y)+f(z)‖≤‖f(x+y+z)‖ in Banach modules over a C∗-algebra, and prove the generalized Hyers-Ulam stability of linear mappings in Banach modules over a C∗-algebra.Choonkil Park© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/908784Some new nonlinear integral inequalities of Gronwall type for retarded functions are established, which extend the results Lipovan (2003) and Pachpatte (2004). These inequalities can be used as basic tools in the study of certain classes of functional differential equations as well as integral equations. A existence and a uniqueness on the solution of the functional differential equation involving several retarded arguments with the initial condition are also indicated.Ravi P. Agarwal, Young-Ho Kim, and S. K. Sen© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/768105We show that the spectral seminorm is useful to study convergence or divergence of vectorial continued fractions in Banach algebras because such convergence or divergence is related to a spectral property.M. Hemdaoui and M. Amzil© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/678014We consider the weakly efficient solution for a class of nonconvex and nonsmooth vector optimization problems in Banach spaces. We show the equivalence between the nonconvex and nonsmooth vector optimization problem and the vector variational-like inequality involving set-valued mappings. We prove some existence results concerned with the weakly efficient solution for the nonconvex and nonsmooth vector optimization problems by using the equivalence and Fan-KKM theorem under some suitable conditions.Zhi-Bin Liu, Jong Kyu Kim, and Nan-Jing Huang© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/381604A second-order nonlinear differential equation, of which some solutions describe the static meniscus free surface (the static liquid bridge free surface between the shaper and the crystal surface) occurring in single crystal ribbon growth, is analyzed. The analysis is focusing on the dependence of the solutions of the equation on the pressure difference p across the free surface. Inequalities are deduced for p, which are necessary or sufficient conditions for the stable and convex free surface of a static meniscus. The obtained results are numerically illustrated in the case of a silicon single crystal ribbon growth. The advantage of these kinds of inequalities is that from them special results can be gleaned concerning the experiment planning and technology design. With this aim this study was undertaken.Stefan Balint and Agneta M. Balint© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/163202We will introduce new means of Cauchy's type Mr,ls(f,μ) defined, for example, as Mr,ls(f,μ)=((l(l−s)/r(r−s))(Mrr(f,μ)−Msr(f,μ)/Mll(f,μ)−Msl(f,μ)))1/(r−l), in the case when l ≠ r ≠ s,  l,r ≠ 0. We will show that this new Cauchy's mean is monotonic, that is, the following result. Theorem. Let t,r,u,v∈ℝ, such that t≤v, r≤u. Then for Mr,ls(f,μ), one has Mt,rs≤Mv,us. We will also give some related comparison results.Matloob Anwar and J. Pečarić© 2008, Hindawi Publishing Corporation. All rights reserved.