Selected Chapters of Geometry, Analysis and Number Theory
(pdf) (ps) (contents-pdf) (460 pages in total)
RGMIA
Monographs
J. Sándor
Selected Chapters of Geometry, Analysis and Number Theory
(pdf) (ps) (contents-pdf)
(460 pages in total)
Preface
The aim of this book is to present short notes or articles,
as well as studies on some topics of Geometry, Analysis, and Number Theory.
The material is divided into ten chapters:
Chapter 1 deals essentially with classical geometric properties of triangles, polygons and tetrahedrons, as well as trigonometric functions.
There are included some recent advances in triangle and tetrahedron inequalities, due to the author.
Chapter 2 studies certain sequences and series of real numbers. Paragraph 2 includes some strange sequences, having interesting definitions
and properties. Other themes introduce the Wallis product, Olivier's criterion,
Dirichlet's beta function, or Bereznai's theorem on the convergence or divergence of infinite series.
The special numbers and sequences of integers of Chapter 3 require special methods of Number theory.
Here some properties of the sequence of primes, of composite numbers, Bernoulli numbers, perfect numbers and generalizations,
or abundant and deficient numbers, are studied. Chapter 4 discusses various algebraic and analytic inequalities,
as the Cauchy-Bunjakovski, Chrystal's, Hadamard's, Jensen's, Chebyshev's,
Fink's, Ky Fan's, Alzer's, etc. inequalities, with many connections and applications to other fields.
Chapter 5 contains 16 notes on the famous Euler gamma function. Many basic as well as new properties (due to the author), including
monotonicity or convexity are studied. Papers 11-13 have been used by many authors in various fields
of study, including Discrete mathematics, Number theory, Analysis, Linear Algebra, etc.
Chapter 6 studies certain new mean value theorems, as well as mean values,
with applications. A special attention is given to applications of Cauchy's mean value theorem,
as well as certain special means, including the so called logarithmic and
identric means, or the Seiffert means. The first theme of Chapter 7 on the Bohr-Mollerup theorem could have been
considered also in Chapter 5 on Euler's gamma function. Here the emphasis is however on functional equations, and along with
the classical gamma function, the $q$-gamma function of Jackson is also involved.
The other themes of this chapter rely on functional equations of more unknowns as Cauchy's, Haruki's and
Cior\u{a}nescu's, Rassias' or Bartha's functional equations. Some functional inequalities are studied, too.
Chapter 8 contains certain Diophantine equations of a new type. The Pell type equations, as well as cubic equations of many unknowns are included.
There are also other interesting equations involving factorials (based on a problem by Smarandache), or arithmetic functions
$\varphi ,\sigma ,\psi $. Paragraph 10 introduces a new concept of Geometry and Number theory:
harmonic triangles. A large chapter involving various arithmetic functions
is Chapter 9. Here many properties of the classical functions as Euler's
totient, the sum and number- of divisors, Jordan's totient, the Smarandache function, etc. are introduced.
There are considered also many new arithmetic functions, as the Euler minimum and maximum functions, the Smarandache minimum and
maximum functions, the star function of an arithmetic function, etc. Finally, Chapter 10 deals with various miscellaneous themes,
as Fermat's "little" theorem and its generalizations, Euler pretty numbers, or certain inequalities of Klamkin, Ky Fan,
or Lehman's, with applications. Article 2 introduces the geometric numbers, not published elsewhere. |