THE POWER MEAN AND THE LOGARITHMIC MEAN

Transcription

1 I ntrnat. J. Mah. Math. Sci. Vol. 5 No. 2 (1982) THE OWER MEAN AND THE LOGARITHMIC MEAN CHRISTOHER OLUTUNDE IMORU Department of Mathematics University of Ife IleIfe Oyo State NIGERIA (Received July 3, 1979 and in revised form, March 24, 1980) ABSTRACT. In a very interesting and recent note, Tungo Lin [I] obtained the least value q and the greatest value p such that M <L<M q is valid for all distinct positive numbers and y where M ( +.y.)s and L s 2 In ln The object of this paper is to give a simpler proof than Lin s of a more general result. More precisely, the author obtained the classes of functions f and agr ha, such that In f (t) h (t) [tl/ + I] > O, t > I. a KE WORDS AND HRASES. Concave Functions, Space of Lebesque integrable functions MATHEMATICS SUBJECT CLASSIFICATION CODES. rimary 26 A INTRODUCTION. There is an etensive literature on the etensions and generalizations of inequalities involving the power mean and the logarithmic mean and their applica

2 338 C.O. IMORU tions [see 25]. Tungo Lin [I] considered the least value p and the greatest value q such that M q <L<M (1.1) is valid for all distinct positive real numbers and y, where L L(,y) y Inlny and H a M.(,y) (X "y +2 c e R are respectively the logarithmic mean and the power mean of order of two distinct positive numbers and y. Indeed, Lin showed that q 0 and p are the best possible values one can get in order that (I.I) may be satisfied for any distinct positive real numbers and y. This result is sharper than an earlier inequality due to Carlson [6], namely: M < L < M for any distinct, y > o (I 2) O As observed by Lin (If], Theroems 3 and 4), if 0<p<I/3, there eist > 0 and y > 0 such that L(,y) > M (,y). On the other hand, there eist > 0 and y > 0 such that L(,y) < M (,y). In fact, if we suppose that > y > o, which is perfectly in order due to the symmetry of L(,y) and M (,y) in and y, one can readily show that L(,y) < Mp(,y), o < p <. (1.3) if is large enough. The reversed inequality holds for smaller values of For instance, numerical calculations with the aid of a computer easily show that the if is greater than (9 9) 4 ( ) s (566188) 6 ( ) 7 ( ) 8 ( ). 9 and ( ) 1 respectively. From these results, we can readily compute the set {(,y); > o and y > o} for which H (,y) < L(,y) > p > For eample, M (,y) < L(,y) on the result (1.3) holds for p,, I0 X set {(,y); < (I )I}. It is interesting to ne that the above esti

3 OWER MEAN AND LOGARITHMIC MEAN 339 mates show that as p o, which is compatible with the lefthand side of in equality (I.I), namely: M (,y) < L(,y) for distinct > y > o. o Furthermore, the critical value of (/y) I/p more than doubles from p 1/(4+n) to p I/(5+n), where n o, I, 2, 5. It is our conjecture that the critical values of (/y)i/ is more than double from p I/n to I/(n+l), n=l, n, The purpose of this note is to give a simpler proof than Carlson or Lin of a more general result. Indeed, we obtain in Section 2 the classes of functions f and h ff e R, for which the function F (t)= In f (t) h a (t) (t I/ + I) t > is strictly positive. This leads to the determination of a polynomial of degree n, (n.>l) which is a lower bound of the function H(t) ()n In t. The comparison of these bounds for n I, 2, with the ones given by Carlson and Lin show that not only are our results more general in form, but they are somewhat sharper. We point out that Carlson s inequalities (1.2) are special cases of an inequality for conve and concave functions. Indeed, from the Jensen inequality for concave functions, t 1 f() d() ff()} dp() t t > O fl dp() fl d() (1.4) where is concave and fg L(R, d), the space of Lebesque p integrable func I/p tions on R, we obtain, on setting f(), p() and () p > or p < o, the inequality In t > [_p tl1/1]p t1 p1 Z > orp < o. (1.5) Since (1.4) is reversed if is conve, inequality (1.5) is reversed if o < p < I. Taing p 2 in (1.5), we obtain the righthand side of inequality (1.2). 2. MAIN RESULTS. We pointed out that Lin, with the aid of the Maclaurin series, showed that

4 340 C.O. IMORU the best possible value of p for which L < M is A proof of this interesting p result without the Maclaurin series was indicated" in the paper Ill by Harley Flanders. Following Flander s approach, we obtain the following result. (z+l) Theorem 2.1: Let F(z) in f(z) ha(z) o R, where fo increasing positive function and ho is differentiable in R. Suppose f(1) is an and ha.(1) 0. Then F (z) > 0, for every z > I, provided o +I hl (z) f (z) h(z) f(z) (z+l) (z+l) + a > o. f (z) f (z) (2.1) roof: Computing the first derivative of f (z) yields Fl(z) Ac(z) (z+l) (+1 Gc(z)] (2.2) where n (z) f f (z) (z) (z+l) a+l and hcl(z) fo(z) h((z) f((z) Ga(z) (z+l) a (z) f f (z) Observe that F(z) 0 if and only if (z+l) a+l GO(z) > 0, z I. Since Fa(1) 0 and Fo(z) z > I. This completes the proof of the theorem. a 8 E (z1) Inz is valid provided a > 0 wherever z > I, we conclude that F (z) > 0, for all Corollary 2.2: For every R, < <, where a +, the inequality c+l < z+l) a, z > 1, (2.3) 1 X+lC [l+ ]]z z a = [{+1 +a ]z+l > 0 (2.4)

5 OWER MEAN AND LOGARITHMIC MEAN 341 a 13 roof" Let h (z) 5 (z _ I) and f (z) z a in Theorem 2 Then the result follows at once. Observe that for all z > I, 13 5 z _ 1) => 2c (za1), provided (13)l<==< (2.5) satisfies the conditions u > 2 and 5 z => o. (2.6) 1) The inequality (2.5) is strict unless u 2 and 13a o, < < u. Consequently, we have 5 wherever (2.4) and (2.6) are valid. (z 1 where z+las well as an upper bound of the function K(z) Corollary 2.3: Suppose condition (2.6) holds. For > y > 0, the inequality O n z n z z1) < (z+l )a a.[ (2.7) Hence for all z > I, the polynomial a(z) {/ 2 a c (13)l<<ot satisfies (2.6), is a lower bound of the function H(z) () In z, : 13 (/a y/u) I/a yl/) y < <, + e, Inlny 2 2(In In y) (2.8) is valid whenever +I 5 [u +1 + =2 B1 C l+] + /ay(a+l)/a 1 a + [u +Iffl31 ffs._.]=l 1/ay (o+ )/c > o (2.9) roof: The result is immediate on setting z _q where > y > o in y inequa ity (2.7).

6 342 C.O. IMORU The cases a 2 and a 3 with a 2 and o # a lead to Carlson s result and Lin s inequality respectively. Suppose a a2a(1+/2) > o; then inequality (2.8) gives an improved lower bound for the power mean H1/a, subject to the condition (2.9). To see this, let us consider the following special cases. Suppose a 1. Then inequality (2.8) yields < y + (1 + < y In Iny In Iny (2 10) 2 provided > y [ (&2+2)]; > o Suppose a 2, we have < (1 +) In Iny z i n Iny (2.11) wherever > y [1++4(2+2)]; > o. If we finally consider the case when 3, then inequality (2.8) yields y In Iny In Iny < (1 + 2 ) y < ( + y (2.12) provided > y(l+ ) + 4(z+2) + 412(z+2)+2(I+)4(+2)]; > o. We pointed out in Section that inequality (1.3) holds for p % if > (9.9) 4 It is pertinent to add here that these values can be obtained by solving for in the inequality (2.9) when 4. To illustrate the usefulness of this method, we compute the set on which inequality (1.3), with p %, is true. Tae 2 o, and 4 in inequality (2.8). The result is y < ( + In Iny 2 (2.13) provided z s 11 z z z 11 z + > o, where z 4 /y. The left hand side of this inequality has the factorization (z + 1) (zl) (z5+246)(z5246).

7 OWER MEAN AND LOGARITHMIC MEAN 343 Consequently, inequality (2.13) is valid provided either > () > I0 or (_Zy ) > Since > y > o, we have L(,y) < M(,y) on the set [(,y); > (9.9)4]. Similar results can be obtained for a 5, 6, 7, Further specializations of the constants 8 < < a, may be of interest in the determination of improved upper and lower bounds for the logarithmic mean and the power mean respectively. REFERENCES I. Tungo Lin: The ower Mean and the Logarithmic Mean; Amer. Math. Monthly 81 (1974), E.B. Leach and M.C. Sholander: Etended Mean Values; Amer. Math. Monthly 85 (1978), E.B. Leach and M.C. Sholander: Corrections to "Etended Mean Values"; Amer. Math. Monthly 85 (1978), K.B. Stolarsy: Generalizations of the Logarithmic Mean; Amer. Math. Mag. 48 (1975), K.B. Stolarsy: The ower and Generalized Logarithmic Mean; Amer. Math. Monthly, to appear. 6. B.C. Carlson: The Logarithmic Mean; Amer. Math. Monthly 79 (1972),

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