Also, we get upper and lower bounds for the hyperpower functionh(x) =xxx

INEQUALITIES ON THE LAMBERT W FUNCTION AND HYPERPOWER FUNCTION

ABDOLHOSSEIN HOORFAR AND MEHDI HASSANI DEPARTMENT OFIRRIGATIONENGINEERING

COLLEGE OFAGRICULTURE

UNIVERSITY OFTEHRAN

P.O. BOX31585-1518, KARAJ, IRAN

hoorfar@ut.ac.ir DEPARTMENT OFMATHEMATICS

INSTITUTE FORADVANCEDSTUDIES INBASICSCIENCES

P.O. BOX45195-1159, ZANJAN, IRAN

mmhassany@member.ams.org

Received 09 April, 2007; accepted 15 March, 2008 Communicated by S.S. Dragomir

ABSTRACT. In this note, we obtain inequalities for the LambertW functionW(x), defined by W(x)e^W(x) = xforx ≥ −e⁻¹. Also, we get upper and lower bounds for the hyperpower functionh(x) =x^xx.

..

.

Key words and phrases: LambertW function, hyperpower function, special function, inequality.

2000 Mathematics Subject Classification. 33E20, 26D07.

1. INTRODUCTION

The LambertW functionW(x), is defined by W(x)e^W(x) = xforx ≥ −e⁻¹. For−e⁻¹ ≤ x < 0, there are two possible values of W(x), which we take values not less than −1. The history of the function goes back to J. H. Lambert (1728-1777). One can find in [2] a more detailed definition ofW as a complex variable function, some historical background and various applications of it in Mathematics and Physics. The expansion

W(x) = logx−log logx+

∞

X

k=0

∞

X

m=1

c_km(log logx)^m (logx)^k+m ,

holds true for large values ofx, with c_km = ⁽⁻¹⁾_m!^kS[k +m, k + 1],where S[k +m, k + 1]is Stirling cycle number [2]. The series in the above expansion is absolutely convergent and it can

107-07

be rearranged into the form W(x) = L₁−L₂+L₂

L₁ + L₂(L₂ −2)

2L²₁ +L₂(2L²₂−9L₂+ 6) 6L³₁ +O

L₂ L₁

4! ,

whereL₁ = logxandL₂ = log logx. Note that bylogwe mean logarithm in the basee. Since the LambertW function appears in some problems in Mathematics, Physics and Engineering, it is very useful to have some explicit bounds for it. In [5] it is shown that

(1.1) logx−log logx < W(x)<logx,

where the left hand side holds true forx > 41.19and the right hand side holds true for x > e.

The aim of the present paper is to obtain some sharper bounds.

2. SOME SHARP BOUNDS FOR THELAMBERTW FUNCTION

It is easy to see that W(−e⁻¹) = −1, W(0) = 0 and W(e) = 1. Also, forx > 0, since W(x)e^W(x)=x >0ande^W(x) >0, we haveW(x)>0. An easy calculation yields that

d

dxW(x) = W(x) x(1 +W(x)).

Thus, x_dx^dW(x) > 0 holds true for x > 0 and consequently W(x) is strictly increasing for x >0(and also for−e⁻¹ ≤x≤0, but not for this reason).

Theorem 2.1. For everyx≥e, we have

(2.1) logx−log logx≤W(x)≤logx− 1

2log logx,

with equality holding only for x = e. The coefficients −1 and−¹₂ of log logx both are best possible for the rangex≥e.

Proof. For the given constant0< p≤2consider the function f(x) = logx− 1

plog logx−W(x), forx≥e. Obviously,

d

dxf(x) = plogx−1−W(x) px(1 +W(x)) logx, and ifp= 2, then

d

dxf(x) = (logx−W(x)) + (logx−1) 2x(1 +W(x)) logx .

Considering the right hand side of (1.1), we have _dx^df(x) > 0 for x > e and consequently f(x) > f(e) = 0, and this gives right hand side of (2.1). Trivially, equality only holds for x=e. If0< p <2, then _dx^df(e) = ^p−2_2ep <0, and this yields that the coefficient−¹₂ oflog logx in the right hand side of (2.1) is the best possible for the rangex≥e.

For the other side, note that logW(x) = logx − W(x) and the inequality logW(x) ≤ log logxholds forx≥e, because of the right hand side of (1.1). Thus,logx−W(x)≤log logx holds for x ≥ e with equality only forx = e. The sharpness of (2.1) with coefficient −1for log logx comes from the relation lim

x→∞(W(x)−logx + log logx) = 0. This completes the

proof.

Now, we try to obtain some upper bounds for the functionW(x)with the main termlogx− log logx. To do this, we need the following lemma.

Lemma 2.2. For everyt∈Randy >0, we have

(t−logy)e^t+y ≥e^t,

with equality fort= logy.

Proof. Lettingf(t) = (t−logy)e^t +y−e^t, we have _dt^df(t) = (t−logy)e^t and _dt^d22f(t) = (t+ 1−logy)e^t. Now, we observe thatf(logy) = _dt^df(logy) = 0and _dt^d22f(logy) = y > 0.

These show that the functionf(t)takes its only minimum value (equal to0) att= logy, which

yields the result of Lemma 2.2.

Theorem 2.3. Fory > ¹_e andx >−¹_e we have

(2.2) W(x)≤log

x+y 1 + logy

,

with equality only forx=ylogy.

Proof. Using the result of Lemma 2.2 witht=W(x), we get (W(x)−logy)e^W(x)−(e^W(x)−y)≥0,

which, consideringW(x)e^W(x) =x,gives(1+logy)e^W(x) ≤x+yand this is desired inequality fory > ¹_e andx >−¹_e. The equality holds whenW(x) = logy, i.e.,x=ylogy.

Corollary 2.4. Forx≥ewe have

(2.3) logx−log logx≤W(x)≤logx−log logx+ log(1 +e⁻¹),

where equality holds in the left hand side forx=eand in the right hand side forx=e^e+1. Proof. Consider (2.2) withy= ^x_e, and the left hand side of (2.1).

Remark 1. Takingy =xin (2.2) we getW(x)≤ logx−log ^1+log₂ ^x

, which is sharper than the right hand side of (2.1).

Theorem 2.5. Forx >1,we have

(2.4) W(x)≥ logx

1 + logx(logx−log logx+ 1), with equality only forx=e.

Proof. Fort >0andx >1, let

f(t) = t−logx

logx −(logt−log logx).

We have _dt^df(t) = _log¹_x − ¹_t and _dt^d22f(t) = _t¹2 >0. Now, we observe that _dt^df(logx) = 0and so mint>0 f(t) = f(logx) = 0.

Thus, fort > 0andx >1we havef(t)≥0,with equality att = logx. Puttingt=W(x)and simplifying, we get the result, with equality atW(x) = logx,or, equivalently, atx=e.

Corollary 2.6. Forx >1we have

W(x)≤(logx)^1+loglogxx.

Proof. This refinement of the right hand side of (1.1) can be obtained by simplifying (2.4) with

W(x) = logx−logW(x).

The bounds we have obtained up to now have the formW(x) = logx−log logx+O(1).

Now, we give bounds with the error termO(^{log log}_logx^x)instead ofO(1).

Theorem 2.7. For everyx≥ewe have (2.5) logx−log logx+1

2

log logx

logx ≤W(x)≤logx−log logx+ e e−1

log logx logx , with equality only forx=e.

Proof. Taking the logarithm of the right hand side of (2.1), we have logW(x)≤log

logx−1

2log logx

= log logx+ log

1−log logx 2 logx

.

UsinglogW(x) = logx−W(x), we get

W(x)≥logx−log logx−log

1−log logx 2 logx

,

which, considering−log(1−t) ≥ tfor0 ≤ t < 1(see [1]) witht = ^{log log}_{2 logx}^x, implies the left hand side of (2.5). To prove the other side, we take the logarithm of the left hand side of (2.1) to get

logW(x)≥log(logx−log logx) = log logx+ log

1− log logx logx

.

Again, usinglogW(x) = logx−W(x), we obtain W(x)≤logx−log logx−log

1−log logx logx

.

Now we use the inequality−log(1−t)≤ _1−t^t for0≤t <1(see [1]) witht= ^{log log}_logx^x, to get

−log

1−log logx logx

≤ log logx logx

1−log logx logx

−1

≤ 1 m

log logx logx , where

m= min

x≥e

1− log logx logx

= 1− 1 e. Thus, we have

−log

1− log logx logx

≤ e e−1

log logx logx ,

which gives the desired bounds. This completes the proof.

3. STUDYING THEHYPERPOWERFUNCTIONh(x) = x^xx.

..

Consider the hyperpower functionh(x) =x^xx.

..

. One can define this function as the limit of the sequence{h_n(x)}_n∈Nwithh₁(x) = xandh_n+1(x) =x^hn(x). It is proven that this sequence converges if and only ife^−e ≤ x≤ e^1e (see [4] and references therein). This function satisfies the relationh(x) = x^h(x), which, on taking the logarithm of both sides and a simple calculation yields

h(x) = W(log(x⁻¹)) log(x⁻¹) .

In this section we obtain some explicit upper and lower bounds for this function. Since the obtained bounds forW(x)hold for large values ofxand since for such values ofxthe value of log(x⁻¹)is negative, we cannot use these bounds to approximateh(x).

Theorem 3.1. Takingλ=e−1−log(e−1) = 1.176956974. . ., fore^−e ≤x≤e^1e we have

(3.1) 1 + log(1−logx)

1−2 logx ≤h(x)≤ λ+ log(1−logx) 1−2 logx ,

where equality holds in the left hand side forx= 1and in the right hand side forx=e^1e. Proof. Fort >0,we havet≥logt+ 1, which takingt =z−logzwithz >0, implies

z

1−2 log(z^1z)

≥log

1−log(z^1z) + 1,

and puttingz^1z =x, or equivalentlyz =h(x), yields thath(x)(1−2 logx)≥log(1−logx)+1;

this is the left hand side (3.1), since1−2 logxis positive fore^−e≤x≤e^1e. Note that equality holds fort=z =x= 1.

For the right hand side, we definef(z) =z−logzwith ¹_e ≤z ≤e. We immediately see that 1 ≤ f(z) ≤ e−1; in fact it takes its minimum value 1atz = 1. Also, consider the function g(t) = logt−t+λ for1 ≤ t ≤ e−1, withλ = e−1−log(e−1). Since _dt^dg(t) = ¹_t −1 andg(e−1) = 0, we obtain the inequalitylogt−t+λ ≥ 0for1 ≤ t ≤ e−1, and putting t=z−logzwith ¹_e ≤z ≤ein this inequality, we obtain

log(1−logz) +λ≥z

1−2 log(z^1z) .

Taking z^1z = x, or equivalentlyz = h(x) yields the right hand side (3.1). Note that equality holds forx=e^1e (z =e, t=e−1). This completes the proof.

REFERENCES

[1] M. ABRAMOWITZ AND I.A. STEGUN, Handbook of Mathematical Functions: with Formu- las, Graphs, and Mathematical Tables, Dover Publications, 1972. [ONLINE: http://www.

convertit.com/Go/ConvertIt/Reference/AMS55.asp].

[2] R.M. CORLESS, G.H. GONNET, D.E.G. HARE, D.J. JEFFREYANDD.E. KNUTH, On the Lam- bertW function, Adv. Comput. Math., 5(4) (1996), 329–359.

[3] R.M. CORLESS, D.J. JEFFREYANDD.E. KNUTH, A sequence of series for the LambertW func- tion, Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation (Kihei, HI), 197-204 (electronic), ACM, New York, 1997.

[4] I. GALIDAKIS AND E.W. WEISSTEIN, "Power Tower." From MathWorld–A Wolfram Web Re- source. [ONLINE:http://mathworld.wolfram.com/PowerTower.html].

[5] M. HASSANI, Approximation of the Lambert W function, RGMIA Research Report Collection, 8(4) (2005), Art. 12.

[6] E.W. WEISSTEIN, Lambert W-Function, from MathWorld–A Wolfram Web Resource. [ONLINE:

http://mathworld.wolfram.com/LambertW-Function.html].