General Mathematics Vol. 12, No. 2 (2004), 47–52
Inequalities for logarithmic
and exponential functions
Eugen Constantinescu
Abstract
In [10] it was announced the precise inequality : for x ≥ 14
!
!
Ã
Ã
12x2 + 6x + 1 − 60x12 +1
12x2 + 6x + 1 − 60x12 +2
1
< < ln
.
ln
x
12x2 − 6x + 1 − 60x12 +2
12x2 − 6x + 1 − 60x12 +1
The aim of the paper is to improve as well as to give a generalization
of above inequalities, see [4].
2000 Mathematical Subject Classification: 26D05,26D07,33B15
Let A(x) := 12x2 + 6x + 1 ,
b(x) =
1
,
B(x) =
1
.
+2
+1
A question raised by Slavko Simić in [10] is to prove following nice inequal60x2
60x2
ities
(1)
ln
µ
A(x) − b(x)
A(−x) − b(x)
¶
1
< < ln
x
µ
A(x) − B(x)
A(−x) − B(x)
¶
,
1
x≥ .
4
Further we need following polynomials
Q3 (x) := 120x3 +60x2 +12x+1 , Q4 (x) := 1680x4 +840x3 +180x2 +20x+1.
47
48
Eugen Constantinescu
Observe that for x ≥ [ 14 , ∞) we have
A(x) − b(x)
Q4 (x)
<
A(−x) − b(x)
Q4 (−x)
In fact
and
A(x) − B(x)
Q3 (x)
=
.
A(−x) − B(x)
−Q3 (−x)
Q4 (x)
A(x) − b(x)
4096x
−
=
where
Q4 (−x) A(−x) − b(x)
C(x)D(x)
C(x) = (4x − 1)2 (1680x2 + 75) + 70(4x − 1) + 11
D(x) = (4x − 1)2 (720x2 + 39) + 30(4x − 1) + 7 .
We shall prove the following ,,better” estimate
µ
µ
¶
¶
Q4 (x)
Q3 (x)
1
1
(2)
ln
< < ln
, x≥ .
Q4 (−x)
x
−Q3 (−x)
4
However, inequalities (2) are equivalent with case n = 2, m = 1 of a more
general result:
µ ¶
n (2n − k)!
(2n)! (−n)k
and let
Theorem 1. Suppose that ck (n) :=
=
k
n!
n! (−2n)k
(3)
Qn (x) :=
n
X
ck (n)xn−k .
k=0
If n, m are positive integers, then for x ≥
1
the following inequalities
2m + 2
are valid
(4)
Q2m+1 (x)
Q2n (x)
< e1/x <
.
Q2n (−x)
−Q2m+1 (−x)
Proof. Note that Qn (x) =
(2n)! n
x + . . . + 1 . The first polynomials Qn are:
n!
Q0 (x) = 1 , Q1 (x) = 2x + 1 , Q2 (x) = 12x2 + 6x + 1
Q3 (x) = 120x3 + 60x2 + 12x + 1.
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
Qn (x) =
Z∞
(−1)n n!
e Pn (1 + 2tx) dt =
(2n)!xn
−t
Z∞
e−t (1 + tx)2n Ln (t) dt
0
0
where Pn is the Legendre polynomial and Ln denotes Laguerre polynomial.
Likewise
(5)
1
Qn (x) =
n!
Z∞
(2n)! n
x 1 F1 (−n ; −2n ; 1/x) .
n!
e−t tn (1 + tx)n dt =
0
Some properties of polynomials Qn are listed in following proposition:
Lemma 1. Let (Qn )n≥1 be the polynomial sequence defined as in (3). Then
for n = 1, 2, ...
• (R ecurrence relation):
• (The roots):
Qn+1 (x) = (4n + 2)xQn (x) + Qn−1 (x) .
Q2n has not real roots. Q2n+1 has only one real root,
namely in (−∞, 0). If Qn (xj ) = 0 , then
1
1
≤ |xk | ≤
,
n(n + 1)
n+1
• (Two identities):
R e(xk ) < 0 , k ∈ {1, 2, ..., n}.
Qn (x)Qn (y) =
n
P
k=0
(6)
(n+k)!
(x
k!(n−k)!
k
+ y) Qk
³


(−1)n Qn (−x)e1/x = Qn (x) + εn (x),



1/x
n Z
(−1)

where εn (x) :=
tn (1 − tx)n et dt.



(2n)!
0
xy
x+y
´
,
50
Eugen Constantinescu
• (A pproximating e1/x ): If Qn (x) 6= 0, then
e1/x =
(7)
Qn (x)
+ rn (x)
(−1)n Qn (−x)
where
εn (x)
= (−1)n e1/x
rn (x) :=
(−1)n Qn (−x)
Z∞
x
e−1/t
dt .
|tQn (−t)|2
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 ≥ 61 the following inequalities hold :
¶
µ
665280x6 + 332640x5 + 75600x4 + 10080x3 + 840x2 + 42x + 1
<
ln
665280x6 − 332640x5 + 75600x4 − 10080x3 + 840x2 − 42x + 1
1
< < ln
x
µ
30240x5 + 15120x4 + 3360x3 + 420x2 + 30x + 1
30240x5 − 15120x4 + 3360x3 − 420x2 + 30x − 1
¶
When x = 1, the recurrence relation from Lemma 1 implies
Corollary 2. If (Yn (λ))∞
n=0 is defined by Y0 (λ) = 1 , Y1 (λ) = λ , and
Yn+1 (λ) = (4n + 2)Yn (λ) + Yn−1 (λ) , n = 1, 2, ...,
then for n large we have
Yn (3)
≈ e. More precisely
Yn (1)
Y2n (3)
Y2m+1 (3)
<e<
, n, m = 0, 1, 2, ...
Y2n (1)
Y2m+1 (1)
Inequalities for logarithmic and exponential functions
51
and
¯
¯
³ ´2n+1
¯
¯
¯e − Yn (3) ¯ < 4 e
, (n ≥ 4).
¯
Yn (1) ¯
4n
Yn (3)
is a rational number. If n = 1000 , then Yn (3) is an
Yn (1)
integer having 3169 digits and Yn (1) is an integer with 3168 digits. Using
Y1000 (3)
a PC, approximation
≈ e gives 6338 correct digits of Napier, s
Y1000 (1)
constant ,, e ”.
Observe that
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, 1953.
[2] Grosswald E., Bessel Polynomials.
Lecture Notes in Mathematics- 698, Springer Verlag, Berlin, 1978.
[3] Hummel P.M., Seebeck C.L., A generalization of T aylor′ s expansion.
Amer.Math.Monthly 56(1949) 243-247.
[4] Lupaş A., Calculul valorilor unor funcţii elementare (I).
Gazeta Matematică (seria A) , VII, 1, (1986) 15-26.
[5] Lupaş A., Numerical Methods.(Romanian), Ed.Constant, Sibiu,2001.
[6] Obreschkoff N., Neue Quadraturformeln. Abhandlugen der Preussischen Akademie der Wissenschaften, (1940), 6-26.
52
Eugen Constantinescu
[7] Obreschkoff N., Verteilung und Berechnung der Nullstellen reeler Polynome. VEB , Band 55 , Berlin, 1963.
[8] Rainville D., Special Functions. MacMillan, New York, 1960.
[9] Saff E.B. , Varga R.S., On the zeros and poles of Padé approximants to
ez (II) . in E.B.Saff & R.S.Varga (eds.), ,, Padé and R ational A pproximation”, Academic Press, New York,pp.195-214, 1976 .
[10] Simić S. , Problem 55,Publikacije Elektrotehnic̆kog Fakulteta, Serija:
Matematika 14(2003) p.112.
,,Lucian Blaga” University of Sibiu
Faculty of Sciences
Department of Mathematics
Str. I.Raţiu 7,
550012-Sibiu, Romania
E-mail: egnconst68@yahoo.com