By József Sándor
This ebook specializes in a few very important classical components of Geometry, research and quantity conception. the fabric is split into ten chapters, together with new advances on triangle or tetrahedral inequalities; targeted sequences and sequence of actual numbers; a number of algebraic or analytic inequalities with functions; distinctive functions(as Euler gamma and beta capabilities) and distinct skill( because the logarithmic, identric, or Seiffert's mean); mathematics services and mathematics inequalities with connections to excellent numbers or comparable fields; and lots of extra. nearly all of the provided themes are in keeping with the unique magazine courses of the writer. This reference paintings can be invaluable for undergraduate collage or students; in addition to lecturers, researchers and professors attracted to those fields of arithmetic.
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Extra resources for Selected Chapters of Geometry, Analysis and Number Theory: Classical Topics in New Perspectives
29) Such simple inequalities have importance in Fourier series. e. x ≤ π − 2 (π − 2 < by π < 4). Thus for π π 2 x ∈ [0, π − 2], relation (29) is better than Jordan’s inequality (1). Applying the same procedure to the cos function and f (x) = −a x − π 2 x+ π , 2 a > 0, 1 since (cos x)x= π = −1, we select a = , and as above one can deduce the 2 π inequality 1 π π π cos x ≥ − x2 + , ∀ x ∈ − , (30) π 4 2 2 The functions sin and cos can be approximated by more complicated functions, too. For example, by taking f (x) = x π 2 − x2 π 2 + x2 47 , one can deduce Redheffer’s inequality (see  or ) sin x π 2 − x2 , ∀ x ∈ [0, π].
For all x ∈ 0, one has 2 3 sin x < x. g. g. , ) were studied by geometric arguments also by Willebrod Snellius (1581-1626), and Christian Huygens (1629-1695). We shall extend de Cusa’s inequality as follows: Theorem 2. Let a, b, c > 0 such that 3b ≤ c ≤ a + b. Then for any π x ∈ 0, one has 2 c sin x < x. a + b cos x 53 π → R, 2 f (x) = x(a + b cos x) − c sin x. Proof. Let us consider the application f : 0, One can remark that f (0) = 0, f (x) = a + b cos x − bx sin x − c cos x, f (0) = a + b − c ≥ 0 by assumption.
5] J. S´andor, On the Jensen-Hadamard inequality, Studia Univ. Babe¸sBolyai, 36(1991), 9-15. 12 On some trigonometric inequalities of Bencze M. 1216, proof of the inequality cos 1 2 n xk ≤ n n k=1 k=1 xk sin xk (1) π , k = 1, 2, . . , n, n ≥ 1. W. Janous  settled this question where xk ∈ 0, 2 in three steps. (i) Inequality (1) is false whenever n ≥ 7. Indeed, let x1 = · · · = xn = 4π . n 4π π 4π 4π Then for n ≥ 9 we have ∈ 0, and (1) becomes ≤ sin in n 2 n n contradiction to the inequality sin x < x valid for all x > 0.
Selected Chapters of Geometry, Analysis and Number Theory: Classical Topics in New Perspectives by József Sándor