Inequalities for logarithmic and exponential functions

Transcription

1 General Mathematics Vol. 12, No. 2 (24), Inequalities for logarithmic and exponential functions Eugen Constantinescu Abstract In [1] it was announced the precise inequality : for x 1 4 ( ) ( ) 12x 2 + 6x x ln x 2 6x < 1 12x 2 x < ln + 6x x x 6x x x 2 +1 The aim of the paper is to improve as well as to give a generalization of above inequalities, see [4]. 2 Mathematical Subject Classification: 26D5,26D7,33B15 Let A(x) := 12x x + 1, b(x) = 6x 2 + 2, B(x) = 1 6x A question raised by Slavko Simić in [1] is to prove following nice inequalities (1) ( ) A(x) b(x) ln < 1 ( ) A(x) B(x) A( x) b(x) x < ln A( x) B(x) Further we need following polynomials, x 1 4. Q 3 (x) := 12x 3 +6x 2 +12x+1, Q 4 (x) := 168x 4 +84x 3 +18x 2 +2x+1. 47

2 48 Eugen Constantinescu Observe that for x [ 1, ) we have 4 A(x) b(x) A( x) b(x) < Q 4(x) Q 4 ( x) and A(x) B(x) A( x) B(x) = Q 3(x) Q 3 ( x). In fact Q 4 (x) A(x) b(x) Q 4 ( x) A( x) b(x) = 496x C(x)D(x) where C(x) = (4x 1) 2 (168x ) + 7(4x 1) + 11 D(x) = (4x 1) 2 (72x ) + 3(4x 1) + 7. We shall prove the following,,better estimate ( ) Q4 (x) ln < 1 ( ) (2) Q 4 ( x) x < ln Q3 (x) Q 3 ( x), x 1 4. However, inequalities (2) are equivalent with case n = 2, m = 1 of a more general result: Theorem 1. Suppose that c k (n) := ( ) n (2n k)! k n! = (2n)! n! ( n) k and let ( 2n) k (3) Q n (x) := n c k (n)x n k. k= If n,m are positive integers, then for x are valid (4) Q 2n (x) Q 2n ( x) < e1/x < 1 2m + 2 Q 2m+1(x) Q 2m+1 ( x). the following inequalities Proof. Note that Q n (x) = (2n)! x n The first polynomials Q n are: n! Q (x) = 1, Q 1 (x) = 2x + 1, Q 2 (x) = 12x 2 + 6x + 1 Q 3 (x) = 12x 3 + 6x x + 1.

3 Inequalities for logarithmic and exponential functions 49 These polynomials are connected to some special polynomials as : Legendre polynomial,laguerre polynomial, Bessel polynomial (see [1],[2],[8] ). For instance, following integral representations are valid Q n (x) = e t P n (1 + 2tx) dt = ( 1)n n! (2n)!x n e t (1 + tx) 2n L n (t) dt where P n is the Legendre polynomial and L n denotes Laguerre polynomial. Likewise (5) Q n (x) = 1 n! e t t n (1 + tx) n dt = (2n)! x n 1F 1 ( n ; 2n ; 1/x). n! Some properties of polynomials Q n are listed in following proposition: Lemma 1. Let (Q n ) n 1 be the polynomial sequence defined as in (3). Then for n = 1, 2,... (Recurrence relation): Q n+1 (x) = (4n + 2)xQ n (x) + Q n 1 (x). (The roots): Q 2n has not real roots. Q 2n+1 has only one real root, namely in (, ). If Q n (x j ) =, then 1 n(n + 1) x k 1 n + 1, Re(x k) <, k {1, 2,...,n}. (Two identities): (6) Q n (x)q n (y) = n ( (n+k)! (x + k!(n k)! y)k xy Q k x+y k= ( 1) n Q n ( x)e 1/x = Q n (x) + ε n (x), 1/x where ε n (x) := ( 1)n t n (1 tx) n e t dt. (2n)! ),

4 5 Eugen Constantinescu (Approximating e 1/x ): If Q n (x), then (7) where r n (x) := e 1/x = Q n (x) ( 1) n Q n ( x) + r n(x) ε n (x) ( 1) n Q n ( x) = ( 1)n e 1/x e 1/t tq n ( t) 2 dt. x The identity (6) is obtained using repeated integrating by parts (see [4]-[5]). Equality (7) is a consequence of (6). The roots are investigated using Eneström -Kakeya theorem (see [5] or N.Obreschkoff, s monograph [7].) Other identities were established using theory of Special Functions[1],[2],[8]. From above Lemma the proof of Theorem is complete (see (6)-(7). In case n = 3, m = 2 from (4) we find Corollary 1. For x 1 the following inequalities hold : 6 ( ) 66528x x x x x x + 1 ln < 66528x x x 4 18x x 2 42x + 1 < 1 ( ) 324x 5 x < ln x x x 2 + 3x x x x 3 42x 2 + 3x 1 When x = 1, the recurrence relation from Lemma 1 implies Corollary 2. If (Y n (λ)) n= is defined by Y (λ) = 1, Y 1 (λ) = λ, and Y n+1 (λ) = (4n + 2)Y n (λ) + Y n 1 (λ), n = 1, 2,..., then for n large we have Y n(3) Y n (1) Y 2n (3) Y 2n (1) < e < Y 2m+1(3) Y 2m+1 (1) e. More precisely, n,m =, 1, 2,...

5 Inequalities for logarithmic and exponential functions 51 and e Y n(3) ( e ) 2n+1 Y n (1) < 4, (n 4). 4n Observe that Y n(3) Y n (1) is a rational number. If n = 1, then Y n(3) is an integer having 3169 digits and Y n (1) is an integer with 3168 digits. Using a PC, approximation Y 1(3) Y 1 (1) e gives 6338 correct digits of Napier, s constant,, e. References [1] Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher Transcendental Functions, vol.i- vol. III= Bateman Manuscript Project. McGraw-Hill, New York, [2] Grosswald E., Bessel Polynomials. Lecture Notes in Mathematics- 698, Springer Verlag, Berlin, [3] Hummel P.M., Seebeck C.L., A generalization of Taylor s expansion. Amer.Math.Monthly 56(1949) [4] Lupaş A., Calculul valorilor unor funcţii elementare (I). Gazeta Matematică (seria A), VII, 1, (1986) [5] Lupaş A., Numerical Methods.(Romanian), Ed.Constant, Sibiu,21. [6] Obreschkoff N., Neue Quadraturformeln. Abhandlugen der Preussischen Akademie der Wissenschaften, (194), 6-26.

6 52 Eugen Constantinescu [7] Obreschkoff N., Verteilung und Berechnung der Nullstellen reeler Polynome. VEB, Band 55, Berlin, [8] Rainville D., Special Functions. MacMillan, New York, 196. [9] Saff E.B., Varga R.S., On the zeros and poles of Padé approximants to e z (II). in E.B.Saff & R.S.Varga (eds.),,, Padé and Rational Approximation, Academic Press, New York,pp , [1] Simić S., Problem 55,Publikacije Elektrotehni ckog Fakulteta, Serija: Matematika 14(23) p.112.,,lucian Blaga University of Sibiu Faculty of Sciences Department of Mathematics Str. I.Raţiu 7, 5512-Sibiu, Romania egnconst68@yahoo.com