Upload
others
View
2
Download
0
Embed Size (px)
Citation preview
Malaysian Journal of Mathematical Sciences 10(1): 23–33 (2016)
MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES
Journal homepage: http://einspem.upm.edu.my/journal
On Inequalities for the Exponential andLogarithmic Functions and Means
Bai-Ni Guo∗1 and Feng Qi2
1School of Mathematics and Informatics, Henan PolytechnicUniversity, Jiaozuo City, Henan Province, 454010, China2Department of Mathematics, College of Science, Tianjin
Polytechnic University, Tianjin City, 300387, China
E-mail: [email protected]∗Corresponding author
ABSTRACT
In the paper, the authors establish a nice inequality for the logarithmicfunction, derive an inequality for the exponential function, and recovera double inequality for bounding the exponential mean in terms of thearithmetic and logarithmic means.
Keywords: inequality, logarithmic function, exponential function, log-arithmic mean, exponential mean.
B.-N. Guo & F. Qi
1. Introduction
In (Kuang, p. 352), two inequalities
e(x+y)/2 <ex − ey
x− y<ex + ey
2
andx+ y
2<
(x− 1)ex − (y − 1)ey
ex − eyfor x 6= y are listed. The first double inequality may be derived easily fromthe Hermite-Hadamard integral inequality for the convex function et on aninterval between x and y. The second one is due to Romanian mathematicianG. Toader, but we do not know its accurate origin. Comparing the aboveinequalities, one may conjecture that
lnex − ey
x− y<
(x− 1)ex − (y − 1)ey
ex − ey< ln
ex + ey
2, x 6= y. (1)
Is the double inequality (1) really valid? The aim of this paper is to answerthe question, to confirm the validity of the double inequality (1), and to find itsequivalence, a double inequality for bounding the exponential mean in termsof the arithmetic and logarithmic means.
2. Main results
We start out from a nice inequality for the logarithmic function.
Theorem 2.1. For a > 0 and a 6= 1, we have
a− 1
ln a
(1 + ln
a− 1
ln a
)< a. (2)
Proof. Let
h(t) = ln t− (t− 1)(1− ln t+ ln |1− t| − ln | ln t|), t 6= 1.
Then straightforward computation gives
h′(t) = ln t− ln |1− t|+ ln | ln t| − 1 +t− 1
t ln t,
h′′(t) =(ln t− t+ 1)(t ln t+ 1− t)
(1− t)t2 ln2 t,
24 Malaysian Journal of Mathematical Sciences
Inequalities for Exponential and Logarithmic Functions
andlimt→1
h(t) = limt→1
h′(t) = 0.
Since1− 1
t< ln t < t− 1, t 6= 1,
it follows that h′′(t) < 0 on (0, 1) and h′′(t) > 0 on (1,∞). These imply that thefirst derivative h′(t) is decreasing on (0, 1), increasing on (1,∞), and positivefor t 6= 1. Furthermore, the function h(t) is increasing for t 6= 0, negative on(0, 1), and positive on (1,∞).
We observe that h(t) ≶ 0 for t > 0 and t 6= 1 are equivalent to
ln t ≶ (t− 1)
(1− ln t+ ln
t− 1
ln t
),
1 >t− 1
ln t
(1− ln t+ ln
t− 1
ln t
),
1
t>
1− 1/t
ln t
(1 + ln
1− 1/t
ln t
),
1
t>
1− 1/t
− ln(1/t)
[1 + ln
1− 1/t
− ln(1/t)
].
Replacing 1t by a in the last inequality leads to the inequality (2). The proof
of Theorem 2.1 is complete.
Theorem 2.2. For x 6= y, we have
lnex − ey
x− y<
(x− 1)ex − (y − 1)ey
ex − ey. (3)
Malaysian Journal of Mathematical Sciences 25
B.-N. Guo & F. Qi
Proof. The inequality (3) may be rewritten as
lnex − ey
x− y<xex − yey
ex − ey− 1,
lnex+1 − ey+1
x− y<xex − yey
ex − ey,
xex − yey
x− y>ex − ey
x− ylnex+1 − ey+1
x− y,∫ y
xet(1 + t) d t
y − x>
∫ y
xet d t
y − xln
∫ y
xet+1 d t
y − x,∫ 1
0
ex+(y−x)t[1 + x+ (y − x)t] d t >∫ 1
0
ex+(y−x)t d t ln
∫ 1
0
ex+(y−x)t+1 d t,∫ 1
0
e(y−x)t[1 + x+ (y − x)t] d t >∫ 1
0
e(y−x)t d t
[x+ 1 + ln
∫ 1
0
e(y−x)t d t
],∫ 1
0
e(y−x)t(y − x)td t >∫ 1
0
e(y−x)t d t ln
∫ 1
0
e(y−x)t d t,
(ln a)
∫ 1
0
attd t >
∫ 1
0
at d t ln
∫ 1
0
at d t,
where a = ey−x. Since the inequality (3) is symmetric between x and y, withoutloss of generality, we assume y > x which means that a > 1. Then, by a directintegration, the last inequality containing the quantity a becomes
1− a+ a ln a
ln a>a− 1
ln alna− 1
ln a,
1− a+ a ln a+ (1− a) ln(a−1ln a
)ln a
> 0,
1− a+ a ln a+ (1− a) ln(a− 1
ln a
)> 0,
a ln a+ (1− a)[1 + ln
(a− 1
ln a
)]> 0,
a ln a > (a− 1)
[1 + ln
(a− 1
ln a
)],
a >a− 1
ln a
[1 + ln
(a− 1
ln a
)].
From the inequality (2) in Theorem 2.1, the inequality (3) follows immediately.The proof of Theorem 2.2 is complete.
26 Malaysian Journal of Mathematical Sciences
Inequalities for Exponential and Logarithmic Functions
Theorem 2.3. For s, t > 0 and s 6= t, we have
s− tln s− ln t
<1
e
(ss
tt
)1/(s−t)
. (4)
Proof. Let ex = s and ey = t. Then the inequality (3) becomes
lns− t
ln s− ln t<
(ln s− 1)s− (ln t− 1)t
s− t
which is equivalent to
s− tln s− ln t
< exp(ln s− 1)s− (ln t− 1)t
s− t=
1
e
(ss
tt
)1/(s−t)
.
The proof of Theorem 2.3 is complete.
Theorem 2.4. For x 6= y, we have
(x− 1)ex − (y − 1)ey
ex − ey< ln
ex + ey
2. (5)
Proof. Since the inequality (5) is symmetric with respect to x and y, withoutloss of generality, we assume y > x. Then the inequality (5) can be rearranged
Malaysian Journal of Mathematical Sciences 27
B.-N. Guo & F. Qi
as
xex − yey
ex − ey< ln
ex+1 + ey+1
2,∫ y
x(1 + t)et d t∫ y
xet d t
< lnex+1 + ey+1
2,∫ 1
0[1 + x+ (y − x)t]ex+(y−x)t d t∫ 1
0ex+(y−x)t d t
< lnex+1 + ey+1
2,
1 + x+
∫ 1
0(y − x)tex+(y−x)t d t∫ 1
0ex+(y−x)t d t
< lnex+1 + ey+1
2,∫ 1
0(y − x)te(y−x)t d t∫ 1
0e(y−x)t d t
< lnex+1 + ey+1
2− (1 + x),∫ 1
0(y − x)te(y−x)t d t∫ 1
0e(y−x)t d t
< lnex+1 + ey+1
2e1+x,∫ 1
0(y − x)te(y−x)t d t∫ 1
0e(y−x)t d t
< ln1 + ey−x
2,
(ln a)∫ 1
0tat d t∫ 1
0at d t
< ln1 + a
2,
(ln a)
∫ 1
0
tat d t <
∫ 1
0
at d t ln1 + a
2,
a ln a+ 1− aln a
<a− 1
ln aln
1 + a
2,
a ln a+ 1− a < (a− 1) ln1 + a
2,
where a = ey−x. Let
H(t) = t ln t+ 1− t− (t− 1) ln1 + t
2, t > 1.
Then a straightforward computation gives
H ′(t) = ln t− t− 1
t+ 1− ln
t+ 1
2and H ′′(t) =
1− tt(t+ 1)2
< 0.
This implies that H ′(t) is decreasing and negative on (1,∞). Furthermore, thefunction H(t) is decreasing and negative on (1,∞). Hence, the inequality (5)is proved. The proof of Theorem 2.4 is complete.
28 Malaysian Journal of Mathematical Sciences
Inequalities for Exponential and Logarithmic Functions
Theorem 2.5. For x 6= y, we have
1
e
(ss
tt
)1/(s−t)
<s+ t
2. (6)
Proof. Taking in (5) ex = s and ey = t figures out
(ln s− 1)s− (ln t− 1)t
s− t< ln
s+ t
2
which is equivalent to
s+ t
2> exp
(ln s− 1)s− (ln t− 1)t
s− t=
1
e
(ss
tt
)1/(s−t)
.
The proof of Theorem 2.5 is complete.
3. Remarks
Remark 3.1. The functions on both sides of the inequality (4) are respectivelycalled the logarithmic and exponential means. See Guo and Qi (2015c), Qiet al. (2014b,c,d, 2015) and plenty of references therein. Therefore, we recovera double inequality for bounding the exponential mean in terms of the arithmeticand logarithmic means.
Remark 3.2. The three inequalities (2), (3), and (4) are equivalent to eachother, although they are of different forms. Similarly, the two inequalities (5)and (6) are also equivalent to each other, although they are of different forms.
Remark 3.3. Taking in (2) a = et for t ∈ R \ {0} yields
et − 1
t
(1 + ln
et − 1
t
)< et,
that is,
lnet − 1
t<
tet
et − 1− 1.
Expanding the functions et−1t and tet
et−1 into power series results in
ln
∞∑k=1
tk−1
k!= ln
∞∑k=0
tk
(k + 1)!<
tet
et − 1− 1 =
∞∑k=1
Bk(1)
k!tk =
∞∑k=1
(−1)kBk
k!tk,
Malaysian Journal of Mathematical Sciences 29
B.-N. Guo & F. Qi
where the Bernoulli polynomials Bk(x) are defined by the generating functions
zexz
ez − 1=
∞∑k=0
Bk(x)
k!zk, |z| < 2π
and the Bernoulli numbers Bk = Bk(0) = (−1)kBk(1).
According to (Comtet, 1974, pp. 140–141, Theorem A), the logarithmic poly-nomials Ln defined by
ln
(1 +
∞∑n=1
gntn
n!
)=
∞∑n=1
Lntn
n!
equal
Ln = Ln(g1, g2, . . . , gn) =
n∑k=1
(−1)k−1(k − 1)!Bn,k(g1, g2, . . . , gn−k+1),
where Bn,k denotes the Bell polynomials of the second kind which are definedby
Bn,k(x1, x2, . . . , xn−k+1) =∑
1≤i≤n,`i∈{0}∪N∑ni=1 i`i=n∑ni=1 `i=k
n!∏n−k+1i=1 `i!
n−k+1∏i=1
(xii!
)`i
for n ≥ k ≥ 0. In (Guo and Qi, 2015a, Theorem 1), the formula
Bn,k
(1
2,1
3, . . . ,
1
n− k + 2
)=
n!
(n+ k)!
k∑`=0
(−1)k−`(n+ k
k − `
)S(n+ `, `)
for n ≥ k ≥ 0 was proved by two approaches, where S(n, k) denotes the Stirlingnumbers of the second kind which may be generated by
(ex − 1)k
k!=
∞∑n=k
S(n, k)xn
n!, k ∈ N.
30 Malaysian Journal of Mathematical Sciences
Inequalities for Exponential and Logarithmic Functions
Consequently, we obtain
lnet − 1
t= ln
∞∑k=0
tk
(k + 1)!= ln
[1 +
∞∑k=1
1
k + 1
tk
k!
]
=
∞∑n=1
[n∑
k=1
(−1)k−1(k − 1)!Bn,k
(1
2,1
3, . . . ,
1
n− k + 2
)]tn
n!
=
∞∑n=1
[n∑
k=1
(−1)k−1(k − 1)!n!
(n+ k)!
k∑`=0
(−1)k−`(n+ k
k − `
)S(n+ `, `)
]tn
n!
=
∞∑n=1
[n∑
k=1
1
k(n+kk
) k∑`=0
(−1)`+1
(n+ k
k − `
)S(n+ `, `)
]tn
n!.
As a result, it follows that
∞∑n=1
[n∑
k=1
1
k(n+kk
) k∑`=0
(−1)`+1
(n+ k
k − `
)S(n+ `, `)
]tn
n!<
∞∑n=1
(−1)nBntn
n!, t 6= 0.
The function ln et−1t may be written as
lnet − 1
t= − ln
t
et − 1= − ln
[1− t
2+
∞∑k=1
B2kt2k
(2k)!
]
= −∞∑
n=1
[n∑
k=1
(−1)k−1(k − 1)!Bn,k
(−1
2, B2, 0, B4, 0, . . . , Bn−k+2
)]tn
n!.
Remark 3.4. For given numbers b > a > 0, let
ga,b(t) =
bt − at
t, t 6= 0;
ln b− ln a, t = 0.(7)
In Guo and Qi (2009b), (Qi et al., 2009, Lemma 1), and (Guo and Qi, 2011,Theorem 2.1), the following conclusions were discovered and applied: the func-tion ga,b(t) is logarithmically convex on (−∞,∞), 3-log-convex on (−∞, 0),and 3-log-concave on (0,∞). For more and detailed information, please referto the survey article Qi et al. (2014a) and closely references therein.
Remark 3.5. In (?, Theorem 5.2), it was obtained that the logarithmic poly-nomials Ln for n ∈ N may be computed by
Ln = gn −n−1∑k=1
(n− 1
k − 1
)gn−kLk (8)
Malaysian Journal of Mathematical Sciences 31
B.-N. Guo & F. Qi
and
Ln = gn + (n− 1)!
n−1∑j=1
(−1)j∑
∑ji=0 mi=n
1≤mk−1≤n−j+k−1−∑k−2
i=0 mi
1≤k≤j
mj
j∏i=0
gmi
mi!. (9)
Remark 3.6. The inequality (2) may also be rewritten as follows. For x > −1and x 6= 0, we have
x
ln(1 + x)
[1 + ln
x
ln(1 + x)
]< 1 + x, (10)
that is,x
(1 + x) ln(1 + x)
[1 + ln
x
ln(1 + x)
]< 1. (11)
The functions xln(1+x) and x
(1+x) ln(1+x) are respectively generating functions ofthe Bernoulli numbers of the second kind and the Cauchy numbers of the secondkind. See Qi (2013, 2014a,b), Qi and Zhang (2015) and closely related referencetherein.
Remark 3.7. In the papers Guo et al. (2008), Guo and Qi (2009a, 2010),Liu et al. (2008), Qi (1997, 2006), Qi et al. (2014a), Zhang et al. (2009),there have been many inequalities related to the exponential function and theirapplications.
Remark 3.8. This paper is a slightly revised version of the preprint Guo andQi (2015b).
Acknowledgements
The authors thanks the mathematicians Ming Li and Zhen-Hang Yang inChina for their valuable comments on the original version of this paper.
References
Comtet, L. (1974). Advanced Combinatorics. D. Reidel Publishing Co., Dor-drecht, enlarged edition. The art of finite and infinite expansions.
Guo, B.-N., Liu, A.-Q., and Qi, F. (2008). Monotonicity and logarithmic con-vexity of three functions involving exponential function. J. Korea Soc. Math.Educ. Ser. B Pure Appl. Math., 15(4):387–392.
32 Malaysian Journal of Mathematical Sciences
Inequalities for Exponential and Logarithmic Functions
Guo, B.-N. and Qi, F. (2009a). Properties and applications of a functioninvolving exponential functions. Commun. Pure Appl. Anal., 8(4):1231–1249.
Guo, B.-N. and Qi, F. (2009b). A simple proof of logarithmic convexity ofextended mean values. Numer. Algorithms, 52(1):89–92.
Guo, B.-N. and Qi, F. (2010). A property of logarithmically absolutely mono-tonic functions and the logarithmically complete monotonicity of a power-exponential function. Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math.Phys., 72(2):21–30.
Guo, B.-N. and Qi, F. (2011). The function (bx−ax)/x: logarithmic convexityand applications to extended mean values. Filomat, 25(4):63–73.
Guo, B.-N. and Qi, F. (2015a). An explicit formula for Bernoulli numbersin terms of Stirling numbers of the second kind. J. Anal. Number Theory,3(1):27–30.
Guo, B.-N. and Qi, F. (2015b). On inequalities for the exponential and loga-rithmic functions and means. ResearchGate Technical Report.
Guo, B.-N. and Qi, F. (2015c). On the degree of the weighted geometric meanas a complete Bernstein function. Afr. Mat., 26(7-8):1253–1262.
Kuang, J.-C. Chángyòng Bùděngshì.
Liu, A.-Q., Li, G.-F., Guo, B.-N., and Qi, F. (2008). Monotonicity and loga-rithmic concavity of two functions involving exponential function. Internat.J. Math. Ed. Sci. Tech., 39(5):686–691.
Qi, F. (1997). A method of constructing inequalities about ex. Univ. Beograd.Publ. Elektrotehn. Fak. Ser. Mat., 8:16–23.
Qi, F. (2006). Three-log-convexity for a class of elementary functions involvingexponential function. J. Math. Anal. Approx. Theory, 1(2):100–103.
Qi, F. (2013). Integral representations and properties of Stirling numbers ofthe first kind. J. Number Theory, 133(7):2307–2319.
Qi, F. (2014a). Explicit formulas for computing Bernoulli numbers of the secondkind and Stirling numbers of the first kind. Filomat, 28(2):319–327.
Qi, F. (2014b). An integral representation, complete monotonicity, and inequal-ities of Cauchy numbers of the second kind. J. Number Theory, 144:244–255.
Qi, F., Cerone, P., Dragomir, S. S., and Srivastava, H. M. (2009). Alternativeproofs for monotonic and logarithmically convex properties of one-parametermean values. Appl. Math. Comput., 208(1):129–133.
Malaysian Journal of Mathematical Sciences 33
B.-N. Guo & F. Qi
Qi, F., Luo, Q.-M., and Guo, B.-N. (2014a). The function (bx − ax)/x: ratio’sproperties. In Analytic number theory, approximation theory, and specialfunctions, pages 485–494. Springer, New York.
Qi, F. and Zhang, X.-J. (2015). An integral representation, some inequalities,and complete monotonicity of the Bernoulli numbers of the second kind.Bull. Korean Math. Soc., 52(3):987–998.
Qi, F., Zhang, X. J., and Li, W. H. (2014b). An integral representation for theweighted geometric mean and its applications. Acta Math. Sin. (Engl. Ser.),30(1):61–68.
Qi, F., Zhang, X.-J., and Li, W.-H. (2014c). Lévy-Khintchine representationof the geometric means of many positive numbers and applications. Math.Inequal. Appl., 17(2):719–729.
Qi, F., Zhang, X.-J., and Li, W.-H. (2014d). Lévy-Khintchine representationsof the weighted geometric mean and the logarithmic mean. Mediterr. J.Math., 11(2):315–327.
Qi, F., Zhang, X.-J., and Li, W.-H. (2015). An elementary proof of the weightedgeometric mean being a Bernstein function. Politehn. Univ. Bucharest Sci.Bull. Ser. A Appl. Math. Phys., 77(1):35–38.
Zhang, S.-Q., Guo, B.-N., and Qi, F. (2009). A concise proof for propertiesof three functions involving the exponential function. Appl. Math. E-Notes,9:177–183.
34 Malaysian Journal of Mathematical Sciences