Monotonicity and Logarithmic Convexity for a Class of Elementary Functions Involving the Exponential Function

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CLASS OF ELEMENTARY FUNCTIONS INVOLVING THE EXPONENTIAL FUNCTION

FENG QI

Abstract. In this paper, the monotonicity and logarithmically convexity of

the function e−₁αt₋−_e−e−tβt are obtained, where t ∈ R and α and β are real

numbers such thatα6=β, (α, β)= (06 ,1) and (α, β)6= (1,0).

1. _Introduction

For real numbersαandβ withα6=β, (α, β)6= (0,1) and (α, β)6= (1,0) and for t∈_R, let

qα,β(t) =  



e−αt₋_e−βt

1−e−t , t6= 0,

β−α, t= 0.

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In order to obtain the best bounds in Gautschi-Kershaw’s inequalities, it was proved in [9] that the functionqα,β(t) is logarithmically convex in (0,∞) and

loga-rithmically concave in (−∞,0) ifβ−α >1 and is logarithmically concave in (0,∞) and logarithmically convex in (−∞,0) if 0< β−α <1.

When ones study the logarithmically completely monotonic property of some functions involving Euler’s gamma Γ function, the psi functionψand the polygamma functions ψ(i) _for _i _∈

N, the elementary function qα,β(t) is encountered now and

then. The so-called logarithmically completely monotonic function on an interval I ⊂_Ris a positive function f which has derivatives of all orders on I and whose logarithm lnf satisfies 0 ≤(−1)k_[ln_f_(x)](k) _<_∞_for _k _∈

Non I. The set of the logarithmically completely monotonic functions onI is denoted byL[I]. For more information on the classL[I], please refer to [1, 2, 3, 5, 6, 7, 8, 9] and the references therein.

The first aim of this paper is to research the monotonicity of the functionqα,β(t).

The first main result of ours is the following Theorem 1 or Corollary 1.

Theorem 1. The following conclusions present the monotonic properties ofqα,β(t).

(1) The functionqα,β(t)is increasing in(0,∞)if either 1≥α+β >2α+ 1or

1≤α+β <2α < α+β+ 1 holds.

(2) The functionqα,β(t)is decreasing in(0,∞)if either1≥α+β >2β+ 1or

1≤α+β <2β < α+β+ 1is valid.

2000Mathematics Subject Classification. Primary 26A48, 26A51; Secondary 26A09, 33B15. Key words and phrases. monotonicity, logarithmically convex function, elementary function, exponential function.

The author was supported in part by the Science Foundation of the Project for Fostering Innovation Talents at Universities of Henan Province, China.

This paper was typeset usingAMS-LA_TEX.

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(3) The functionqα,β(t)is increasing in(−∞,0) if either2α > α+β+ 1≥2

orα+β <2β < α+β+ 1≤2 validates.

(4) The functionqα,β(t)is decreasing in (−∞,0)if either 2β > α+β+ 1≥2

orα+β <2α < α+β+ 1≤2 sounds.

(5) The function qα,β(t) is increasing in (−∞,∞) if and only if one of the

following conditions holds: (a) α=β+ 1>1, (b) α > β+ 1≥1, (c) β=α+ 1<1, (d) 1≥β > α+ 1,

(e) α < β < α+ 1≤1,

(f) β+ 1≤α+β <2α < α+β+ 1.

(6) The function qα,β(t) is decreasing in (−∞,∞) if and only if one of the

following conditions holds: (a) β=α+ 1>1, (b) β > α+ 1≥1,

(c) β < α < β+ 1≤1, (d) 1> α=β+ 1,

(e) 1≥α > β+ 1,

(f) α+ 1≤α+β <2β < α+β+ 1.

Remark 1. The (α, β)-domain where the functionqα,β(t) is monotonic in Theorem 1

can be described respectively by Figure 1 to Figure 6 below.

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` ` `` ` ` ` ` `` ` `

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β=α

β=α+ 1

β=α−1

β= 1−α

Figure 1. (α, β)-domain where the function qα,β(t) is increasing

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` ` `` ` ` ` ` `` ` `

` ` ``

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β =α β =α+ 1

β =α−1 β= 1−α

Figure 2. (α, β)-domain where the function qα,β(t) is decreasing

in (0,∞) in Theorem 1

Remark 2. Note that the (α, β)-domain where the functionqα,β(t) is increasing (or

decreasing) in (0,∞) (or in (−∞,0)) is an union where the functionqα,β(t) increases

(or decreases) in either (0,∞) (or (−∞,0)) or (−∞,∞). Therefore, Theorem 1 can be restated as the following Corollary 1.

Corollary 1. The following conclusions describe the monotonic properties ofqα,β(t).

(1) The function qα,β(t) is increasing in (−∞,∞) if and only if (α, β) ∈ {(α, β) : α > β ≥ 0, α ≥ 1} ∪ {(α, β) : α < β ≤ 0} ∪ {(α, β) : α ≤

β−1,0≤β≤1} \ {(1,0),(0,1)}.

(2) The function qα,β(t) is decreasing in (−∞,∞) if and only if (α, β) ∈ {(α, β) : β > α ≥ 0, β ≥ 1} ∪ {(α, β) : β < α ≤ 0} ∪ {(α, β) : β ≤

α−1,0≤α≤1} \ {(1,0),(0,1)}.

(3) The functionqα,β(t)is increasing in (0,∞)if and only if(α, β)∈ {(α, β) :

α > β ≥ 1

2} ∪ {(α, β) : α ≥1−β,0 ≤ β < 1

2} ∪ {(α, β) : α+ 1 ≤ β ≤

1−α, α <0} ∪ {(α, β) :β−1≤α < β≤0} \ {(1,0)}.

(4) The functionqα,β(t)is decreasing in(0,∞)if and only if(α, β)∈ {(α, β) :

β≥1−α,1₂ > α≥0}∪{(α, β) :β > α≥ 1

2}∪{(α, β) :β < α≤0}∪{(α, β) :

β≤α−1,0≤α≤1} ∪ {(α, β) : 1≤α≤1−β} \ {(1,0),(0,1)}.

(5) The functionqα,β(t)is increasing in(−∞,0)if and only if(α, β)∈ {(α, β) :

1−α≤β < α, α≥1} ∪ {(α, β) : α < β ≤1, α≤0} ∪ {(α, β) :α < β ≤

1−α,0≤α < 1

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` ` `` β=α

β = 1−α

β=α−1

β =α+ 1

in (−∞,0) in Theorem 1

(6) The functionqα,β(t)is decreasing in(−∞,0)if and only if(α, β)∈ {(α, β) :

1−β≤α < β, β≥1} ∪ {(α, β) :β < α≤ 1

2} ∪ {(α, β) :β≤1−α, 1 2 < α≤

1} \ {(1,0),(0,1)}.

Remark 3. The corresponding (α, β)-domains where the function qα,β(t) is

mono-tonic in Corollary 1 can be described respectively by Figure 5 to Figure 10 below.

The second aim of this paper is to reconsider the logarithmically convexity of the functionqα,β(t) by a very simpler approach than that in [9]. The second main

result of ours is the following Theorem 2.

Theorem 2. The functionqα,β(t)in(−∞,∞)is logarithmically convex ifβ−α >1

and logarithmically concave if 0< β−α <1.

Remark 4. Theorem 2 shows that the logarithmically convexity and logarithmically concavity in the interval (−∞,0) ofqα,β(t) presented in [9] and mentioned at the

beginning of this paper are wrong. However, this does not affect the correctness of the main results established in [9], since the wrong properties aboutqα,β(t) in the

interval (−∞,0) are unuseful there luckily.

Remark 5. Recall that ar-times differentiable functionf(x)>0 is said to be r-log-convex (orr-log-concave) on an intervalIwithr≥2 if and only if [lnf(x)](r)_exists

and [lnf(x)](r) _≥_{0 (or [ln}_f_(x)](r)_≤_{0) on} _{I. In [4], the following conclusions are}

obtained: If 1> β−α >0, thenqα,β(t) is 3-log-convex in (0,∞) and 3-log-concave

in (−∞,0); ifβ−α >1, thenqα,β(t) is 3-log-concave in (0,∞) and 3-log-convex in

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β =α

β= 1−α β =α−1

β=α+ 1

2. Proofs of theorems

Proof of Theorem 1. It is clear that the functionqα,β(t) can be rewritten as

qα,β(t) =

sinh(β−α₂ )t sinh₂t exp

(1−α−β)t 2 ,pα,β

_t

2

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If α=β + 1, then qα,β(t) = −e−βt is increasing forβ >0 and decreasing for

β <0 in (−∞,∞). If α=β−1, thenqα,β(t) =e−αt is decreasing for α >0 and

increasing forα <0 in (−∞,∞).

For|α−β| 6= 1, direct differentiation shows

p0_α,β(t) = sinh((β−α)t) sinht e

(1−α−β)t_ϕ α,β(t),

where

ϕα,β(t) = (β−α) coth((β−α)t)−cotht−α−β+ 1 (3)

and

ϕ0_α,β(t) =

₁

sinht

2 −

_β₋_α

sinh((β−α)t)

2

= 1 t2

_t

sinht

2 −

_(β₋_α)t

sinh((β−α)t)

2

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β=α β=α−1 β=α+ 1

``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

in (−∞,∞) in Theorem 1 and Corollary 1

Since ϕ0_α,β(t) = ϕ0_α,β(−t) and the function _sinht _t > 0 is decreasing in (0,∞) and increasing in (−∞,0), thenϕ0_α,β(t)≥ 0 for |α−β| >1 and ϕ0_α,β(t) ≤ 0 for 0 <|α−β| <1 in (−∞,∞). This means that the functionϕα,β(t) is increasing

for|α−β|>1 and decreasing for 0<|α−β|<1 in (−∞,∞). It is not difficult to obtain limt→−∞ϕα,β(t) = 2−α−β− |α−β|, limt→0ϕα,β(t) = 1−α−β and

limt→∞ϕα,β(t) =|α−β| −α−β.

1.If β > α+ 1, thenβ−α > 0,|α−β|>1, limt→−∞ϕα,β(t) = 2(1−β) and

limt→∞ϕα,β(t) = −2α. Further, ifα ≥ 0, then ϕα,β(t) < 0 and p0α,β(t) < 0 in

(−∞,∞), and thenpα,β(t) and qα,β(t) are decreasing in (−∞,∞). Therefore, for

β > α+ 1≥1, the functionqα,β(t) is decreasing in (−∞,∞).

Ifβ > α+ 1 andβ ≤1, then limt→−∞ϕα,β(t)≥0,ϕα,β(t)>0 andp0α,β(t)>0

in (−∞,∞), and then pα,β(t) and qα,β(t) are increasing in (−∞,∞). Hence, for

1≥β > α+ 1, the functionqα,β(t) is increasing in (−∞,∞).

Ifβ > α+ 1 andα+β≤1, then limt→0ϕα,β(t)≥0,ϕα,β(t)>0 andp0α,β(t)>0

in (0,∞), and thenpα,β(t) andqα,β(t) are increasing in (0,∞). Consequently, for

2α+ 1< α+β≤1, the functionqα,β(t) is increasing in (0,∞).

Ifβ > α+ 1 andα+β≥1, then limt→0ϕα,β(t)≤0,ϕα,β(t)<0 andp0α,β(t)<0

in (−∞,0), and thenpα,β(t) andqα,β(t) are decreasing in (−∞,0). Therefore, for

2β > α+β+ 1≥2, the functionqα,β(t) is decreasing in (−∞,0).

2.If α < β < α+ 1, then β−α > 0 and |α−β|<1. Further, if α≤0, then ϕα,β(t)>0 andp0α,β(t)>0 in (−∞,∞), and thenpα,β(t) andqα,β(t) are increasing

in (−∞,∞). Accordingly, forα < β < α+ 1≤1, the functionqα,β(t) is increasing

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β =α

β =α−1

β=α+ 1

``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` `` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

in (−∞,∞) in Theorem 1 and Corollary 1

If α < β < α+ 1 and β ≥ 1, then limt→−∞ϕα,β(t) ≤ 0, ϕα,β(t) < 0 and

p0_α,β(t)<0 in (−∞,∞), and then pα,β(t) and qα,β(t) are decreasing in (−∞,∞).

Therefore, for α+ 1≤α+β <2β < α+β+ 1, the functionqα,β(t) is decreasing

in (−∞,∞).

If α < β < α+ 1 and α+β ≤ 1, then limt→0ϕα,β(t) ≥ 0, ϕα,β(t) > 0 and

p0_α,β(t)>0 in (−∞,0), and thenpα,β(t) andqα,β(t) are increasing in (−∞,0). As

a result, for α+β < 2β < α+β+ 1 ≤ 2, the function qα,β(t) is increasing in

(−∞,0).

If α < β < α+ 1 and α+β ≥ 1, then limt→0ϕα,β(t) ≤ 0, ϕα,β(t) < 0 and

p0_α,β(t)<0 in (0,∞), and thenpα,β(t) and qα,β(t) are decreasing in (0,∞).

Con-sequently, for 1 ≤α+β <2β < α+β+ 1, the functionqα,β(t) is decreasing in

(0,∞).

3.If α > β+ 1, thenβ−α <0, |α−β|>1, limt→−∞ϕα,β(t) = 2(1−α) and

limt→∞ϕα,β(t) = −2β. Further, if β ≥ 0, then ϕα,β(t) < 0 and p0α,β(t) > 0 in

(−∞,∞), and then pα,β(t) and qα,β(t) are increasing in (−∞,∞). Therefore, for

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β=α+ 1

Ifα > β+ 1 andα≤1, then limt→−∞ϕα,β(t)≥0,ϕα,β(t)>0 andp0α,β(t)<0

in (−∞,∞), and then pα,β(t) andqα,β(t) are decreasing in (−∞,∞). Hence, for

1≥α > β+ 1, the functionqα,β(t) is decreasing in (−∞,∞).

Ifα > β+ 1 andα+β≤1, then limt→0ϕα,β(t)≥0,ϕα,β(t)>0 andp0α,β(t)<0

in (0,∞), and then pα,β(t) andqα,β(t) are decreasing in (0,∞). Accordingly, for

1≥α+β >2β+ 1, the functionqα,β(t) is decreasing in (0,∞).

Ifα > β+ 1 andα+β≥1, then limt→0ϕα,β(t)≤0,ϕα,β(t)<0 andp0α,β(t)>0

in (−∞,0), and then pα,β(t) and qα,β(t) are increasing in (−∞,0). Hence, for

2α > α+β+ 1≥2, the functionqα,β(t) is increasing in (−∞,0).

4.If β < α < β+ 1, then β−α <0 and |α−β|<1. Further, if β ≤0, then ϕα,β(t)>0 andp0α,β(t)<0 in (−∞,∞), and thenpα,β(t) andqα,β(t) are decreasing

in (−∞,∞). Therefore, for β < α < β+ 1≤1, the function qα,β(t) is decreasing

in (−∞,∞).

If β < α < β + 1 and α ≥ 1, then limt→−∞ϕα,β(t) ≤ 0, ϕα,β(t) < 0 and

p0_α,β(t)>0 in (−∞,∞), and then pα,β(t) andqα,β(t) are increasing in (−∞,∞).

Accordingly, forβ+ 1≤α+β <2α < α+β+ 1, the functionqα,β(t) is increasing

in (−∞,∞).

If β < α < β+ 1 and α+β ≤ 1, then limt→0ϕα,β(t) ≥ 0, ϕα,β(t) > 0 and

p0_α,β(t) < 0 in (−∞,0), and then pα,β(t) and qα,β(t) are decreasing in (−∞,0).

Consequently, forα+β <2α < α+β+ 1≤2, the functionqα,β(t) is decreasing in

(−∞,0).

If β < α < β+ 1 and α+β ≥ 1, then limt→0ϕα,β(t) ≤ 0, ϕα,β(t) < 0 and

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β =α

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result, for 1≤α+β <2α < α+β+ 1, the functionqα,β(t) is increasing in (0,∞).

The proof of Theorem 1 is complete.

Proof of Theorem 2. For β > α, the functions qα,β(t) andpα,β(t), related by (2),

are positive. Taking logarithm ofpα,β(t) and differentiating yields

lnpα,β(t) = ln sinh((β−α)t)−ln sinht+ (1−α−β)t,

[lnpα,β(t)]0= (β−α) coth((β−α)t)−cotht−α−β+ 1 =ϕα,β(t),

whereϕα,β(t) is defined by (3).

By the same argument as in the proof of Theorem 1 on page 5, it is easy to see thatϕ0_α,β(t) = [lnpα,β(t)]00 ≥0 forβ−α >1 and ϕ0α,β(t) = [pα,β(t)]00 ≤0 for

0 < β−α < 1 in (−∞,∞). This means that the function pα,β(t) = qα,β(2t) is

logarithmically convex forβ−α >1 and logarithmically concave for 0< β−α <1 in the whole axis (−∞,∞). The proof of Theorem 2 is complete.

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β=α β=α−1 β =α+ 1

``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

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` ` `` ` ` ``

` ` ``_{` ` `}

β = 1−α

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andq-gamma functions, Proc. Amer. Math. Soc.134(2006), no. 4, 1153–1160.

[3] F. Qi, Certain logarithmically N-alternating monotonic functions involving gamma and

q-gamma functions, RGMIA Res. Rep. Coll. 8 (2005), no. 3, Art. 5. Available online at

http://rgmia.vu.edu.au/v8n3.html.

[4] F. Qi,Three-log-convexity for a class of elementary functions involving exponential function,

J. Math. Anal. Approx. Theory1(2006), no. 2, in press.

[5] F. Qi and Ch.-P. Chen,A complete monotonicity property of the gamma function, J. Math.

Anal. Appl.296(2004), no. 2, 603–607.

[6] F. Qi and B.-N. Guo,Complete monotonicities of functions involving the gamma and digamma

functions, RGMIA Res. Rep. Coll.7 (2004), no. 1, Art. 8, 63–72. Available online athttp: //rgmia.vu.edu.au/v7n1.html.

[7] F. Qi, B.-N. Guo, and Ch.-P. Chen,Some completely monotonic functions involving the gamma

and polygamma functions, RGMIA Res. Rep. Coll.7(2004), no. 1, Art. 5, 31–36. Available

online athttp://rgmia.vu.edu.au/v7n1.html.

[8] F. Qi, B.-N. Guo, and Ch.-P. Chen,Some completely monotonic functions involving the gamma

and polygamma functions, J. Austral. Math. Soc.80(2006), 81–88.

[9] F. Qi, B.-N. Guo, and Ch.-P. Chen,The best bounds in Gautschi-Kershaw inequalities, Math.

Inequal. Appl. (2006), in press. RGMIA Res. Rep. Coll.8(2005), no. 2, Art. 17. Available

online athttp://rgmia.vu.edu.au/v8n2.html.

(F. Qi)Research Institute of Mathematical Inequality Theory, Henan Polytechnic

University, Jiaozuo City, Henan Province, 454010, China

E-mail address: [email protected], [email protected], [email protected], [email protected], [email protected]

(11)

-α

1 O

6

β

1

` ` `` ` ` ` ` `` ` `

` ` `

β=α

β =α−1

β =α+ 1

``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` ``` `` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `@

@ @ @ @

@ @ @ @ @

@

@ @ @

@ @ ` ` ``

` ` `` ` ` ``

` ` `` ` ` ``_{` ` `}

β= 1−α

Figure 10. (α, β)-domain where the functionqα,β(t) is decreasing