József Sándor

Handbook of number theory II

Preface. Basic Symbols. Basic Notations. I. Eulers phi-function. II. The arithmetical function d(n), its generalizations and its analogues. III. Sum-of-divisors function, generalizations, analogues Perfect numbers and related problems. IV. P, p, B, beta and related functions. V. omega(n), Omega(n) and related functions. VI. Function mu k-free and k-full numbers. VII. Functions pi(x), psi(x), theta(x), and the sequence of prime numbers. VIII. Primes in arithmetic progressions and other sequences. IX. Additive and diophantine problems involving primes. X. Exponential sums. XI. Character sums. XII. Binomial coefficients, consecutive integers and related problems. XIII. Estimates involving finite groups and semi-simple rings. XIV. Partitions. XV. Congruences, residues and primitive roots. XVI. Additive and multiplicative functions. Index of authors.

On the identric and logarithmic means

SummaryLeta, b > 0 be positive real numbers. The identric meanI(a, b) of a andb is defined byI = I(a, b) = (1/e)(bb/aa)1/(b−a), fora ≠ b, I(a, a) = a; while the logarithmic meanL(a, b) ofa andb isL = L(a, b) = (b − a)/(logb − loga), fora ≠ b, L(a, a) = a. Let us denote the arithmetic mean ofa andb byA = A(a, b) = (a + b)/2 and the geometric mean byG =G(a, b) =

Two inequalities for means

\sqrt {ab}

Inequalities for hyperbolic functions

. In this paper we obtain some improvements of known results and new inequalities containing the identric and logarithmic means. The material is divided into six parts. Section 1 contains a review of the most important results which are known for the above means. In Section 2 we prove an inequality which leads to some improvements of known inequalities. Section 3 gives an application of monotonic functions having a logarithmically convex (or concave) inverse function. Section 4 works with the logarithm ofI(a, b), while Section 5 is based on the integral representation of means and related integral inequalities. Finally, Section 6 suggests a new mean and certain generalizations of the identric and logarithmic means.

On Bessel's and Gram's inequalities in prehilbertian spaces

We prove certain new inequalities for special means of two arguments, includ- ing the identric, arithmetic, and geometric means.

Inequalities involving Jacobian elliptic functions and their inverses

We prove two new inequalities for the identric mean and a mean related to the arithmetic and geometric mean of two numbers.

On an Arithmetic Inequality

Refinements of the inequalities of Ky Fan [3], Wang and Wang [16], Sándor and Trif [12], and Sándor [14] are obtained. Generalizations and new proofs of some of these inequalities are also included. Mathematics subject classification (2000): 26D15, 26D99.

On some exponential means. Part II

Abstract Several inequalities involving hyperbolic functions are derived. Some of them are obtained with the aid of Stolarsky and Gini means.