- Email: [email protected]
On a Ky Fan inequality of the complementary A-G type and its variants
JOURNAL
OF hfATHEMATICAL
ANALYSIS
AND
APPLICATIONS
73, 501-505 (I 980)
On a Ky Fan inequality of the Complementary A-G Type and Its Variants CHUNG-LIE WANG* Department of Mathematics and Statistics, University of Regina, Saskatchewan S4S OA2, Canada Submitted by E. S. Lee
I. INTRODUCTION .~ND SUMMARY
In Beckenbach and Bellman [l] (see also [5, p. 363]), the following published inequality due to Ky Fan is recorded:
un-
KY FAN THEOREM [l, p. 51. For xj E (0, &), j = l,..., n, we have
(l-I %)/(C %)” G (II-l (1 - Xi,)/@
(1 - Xj))”
(1)
with equality only if all X~are equal. Here and in what follows x and n are used to indicate x:,‘=, and ny=, , yespeetively, whenever confusion is unlikely to OCCUY. In Levinson [3], inequality (1) is extended as follows: N. LEVINSON THEOREM [3, p. 1331. Let 4 have a nonnegative third derivative MI (0, 2b). Then for X~E (0, b] and p, > 0, j = l,..., n, with P, = C p, , we have
Moreover if d3$(u)/du3 > 0 on (0, 2b) then equality are equal.
occurs
above only if all xi
In Popoviciu [4] or Bullen [2], inequality (2) is again generalized as follows: * This paper was written while the author was on sabbatical leave at the University of Delaware for the year 1978-1979. The author was supported in part by the NSERC of Canada (Grant A3116).
501 0022-247X/80/020315-23$02.00/0 Copyright Q 1980 by Academic Press, Inc. .All rights of reproduction in any form reserved.
502
CHUNG-LIE
WANG
T. POPOVICIU THEOREM [2, p. 1101. (a) Let 4 be a real-aalued function on [a, b] and x, , y, (j = I,..., n)2n points on [a, b] such that max{xl
thenifp,>O,j=l,...,
,.. ., 4
< minh
,...,m),
xl +
3-conerex
y1 = -** = x, + yn;
(3)
n,P,=xpp,,
If CJis strictly 3-convex, then equality occurs in (4) only af all x, are equal. (b) If for a continuous function 4 inequality (4) holds (strictly) for every n and all 2n (distinct) points satisfying (3) and all pi > 0, then 4 is (strictly) 3conzfex. An inequality converse to (4) can be found in Vasic and JaniC [7]. In view of the above, the Levinson inequality (2) which originated from the Ky Fan inequality (l), is well developed. On the other hand, the development of the remarkable Ky Fan inequality (1) came to a standstill after the elegant work of Levinson. It should be noted that, as indicated in [ 1, p. 51, inequality (1) was originally established by forward and backward induction as often used for establishing the A-G (abbreviation of arithmetic and geometric) inequality [I, p. 41. In fact, inequality (1) is a replica of the A-G inequality (see also below). It appears that the intention of Beckenbach and Bellman [I] is to display a strong resemblance between the Ky Fan inequality (1) and the A-G inequality. Since there are 12 proofs of the A-G inequality given in [I, Sects. 5161 alone, it is natural to ask the question: “Are there more proofs of inequality (1) in addition to the one by Levinson [3] and the original, unpublished one ?” In this note, we will give an affirmative answer to the above question. In Section 2, we will rewrite (I) in a little more general form (Eq. (5)) and call it a Ky Fan inequality of the complementary A-G type. We will also present two variants of (5) and an inequality of Rado type (see, e.g., [I, p. 12; 4, p. 61; 5, p. 941) related to its variant (Eq. (6)). In Section 3, we will conclude with a conjecture associated with this new development of (1) and some remarks.
2. MAIN
RESULTS
Let p =z (p, ,..., p,J and .x = (x1 ,..., x,) be two sequences of positive numbers such that s, E (0, 41, j = l,..., n. Let A, , Gk (resp. ,4; and Gk) designate
ON
A KY FAN
503
INEQUALITY
(resp. complementary) arithmetic and geometric means of the numbers x1 )..., xl: (resp. 1 - x1 ,..., 1 - xJ with unequal weights pi ,..., p, , i.e.,
-4, = A&;
p) = : p,s,/Pk )
G, = G,(x; p) = fi xf’+ 3=1
,=l
G; = G,( I - .v; p),
,-J;. = &( I - x; p), where P, = x:=, p, , k = 2 ,..., n. We now give a Ky Fan inequality
of complementary
A-G type as follows:
which is equivalent to (1) (if p, = 1/n for all j). Since A, + .4:, = 1, (5) is equivalent to
G, d AdG, + GJ
(6)
A;(G,
(7)
or
Moreover,
+ G)
< ‘X .
since G, < GL and A, < -4; , (6) and (7) yield
Inequalities (6) and (7) are two variants of (5). Because of their equivalence, it is necessary to prove only one of them. It is also very interesting and significant to note that, like the A-G inequality, the Ky Fan inequality of complementary A-G type possesses an inequality of Rado type (see, e.g., [I, p. 121) related to (6):
PnL%(G,+ ‘$3 - 61 3 f’,d&-dG,-1
+ G-d - Gn-,I.
(9)
(Note also that an inequality of Rado type related to (7) can be given likewise.) Naturally, the establishment of (6) follows inductively from (9). To establish (9) consider the function f defined by
f (.s) = 4(G, + G;) - G, . Since A, = (P,-l/P,)A,-l
_
GP,-IIP+,IP,. 11-I n
+ (p,/P,)xn
, G, = Gi:yJPnx$IPm, etc. We have
504
CHUNG-LIE WANG
Simple manipulations reveal that
and
where
and
Since it is easy to see that U, > 0 always (unless Xj = + , j = l,..., n), the minimum is attained for
at which Vn = 0, W,, > 0, d2f/dx,2 > 0, and so
f(Xn) >f(G) = +
n
(Gnel+
G~-l)-P,‘Pn[tln-l(G,-~
+ G-I) - G-,1- (10)
By use of the fact that G, + GI, < 1, k = l,..., n, (9) follows from (10) immediately.
3. REMARKS AND A CONJECTURE From the above results, it may be expected that (6) can be established by some other standard approaches used for establishing the A-G inequality (see, e.g. [l, 4, 51). However, it should be pointed out here that several unsuccessful attempts have been made to establish (6) by directly using some approaches given in [I, Sects. 5-161. For this reason, it appears necessary to modify somewhat the standard approaches used for establishing the A-G inequality in establishing (6).
ON A KY FAN INEQUALITY
505
Conjecture. Let C$be an increasing function satisfying all conditions assumed in Levinson [3] (see above). Then we have
Obviously, (6) is the case 4(u) = log u and b = 8 of (I 1). REFERENCES New 1. E. F. BECKENBACH AND R. BELLRIAN, “Inequalities, ” 2nd ed., Springer-Verlag, York, 1965. 2. P. S. BULLEN, An inequality of N. Levinson, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., Nos. 412-460 (1973), 109-112. 3. N. LEVINSON, Generalization of an inequality of Ky Fan, J. Math. Anal. Appl. 8 (1964), 133-134. 4. G. H. HARDY, J. E. LITTLEWOOD AND G. P~LYA, “Inequalities,” 2nd. ed., Cambridge Univ. Press, Cambridge, 1952. 5. D. S. MITRINOVIC, “Analytic Inequalities,” Springer-Verlag, Berlin, 1970. 6. T. POPO~ICIU, Sur une inegalitb de N. Levinson, Mathematics (CZuj) 6 (1964), 301-
306. 7. P. M. VA& AND R. R. JANIC, On an inequality Belgrade 10 (1970), 155-157.
of N. Levinson,
Publ. Inst. Math.
OF hfATHEMATICAL
ANALYSIS
AND
APPLICATIONS
73, 501-505 (I 980)
On a Ky Fan inequality of the Complementary A-G Type and Its Variants CHUNG-LIE WANG* Department of Mathematics and Statistics, University of Regina, Saskatchewan S4S OA2, Canada Submitted by E. S. Lee
I. INTRODUCTION .~ND SUMMARY
In Beckenbach and Bellman [l] (see also [5, p. 363]), the following published inequality due to Ky Fan is recorded:
un-
KY FAN THEOREM [l, p. 51. For xj E (0, &), j = l,..., n, we have
(l-I %)/(C %)” G (II-l (1 - Xi,)/@
(1 - Xj))”
(1)
with equality only if all X~are equal. Here and in what follows x and n are used to indicate x:,‘=, and ny=, , yespeetively, whenever confusion is unlikely to OCCUY. In Levinson [3], inequality (1) is extended as follows: N. LEVINSON THEOREM [3, p. 1331. Let 4 have a nonnegative third derivative MI (0, 2b). Then for X~E (0, b] and p, > 0, j = l,..., n, with P, = C p, , we have
Moreover if d3$(u)/du3 > 0 on (0, 2b) then equality are equal.
occurs
above only if all xi
In Popoviciu [4] or Bullen [2], inequality (2) is again generalized as follows: * This paper was written while the author was on sabbatical leave at the University of Delaware for the year 1978-1979. The author was supported in part by the NSERC of Canada (Grant A3116).
501 0022-247X/80/020315-23$02.00/0 Copyright Q 1980 by Academic Press, Inc. .All rights of reproduction in any form reserved.
502
CHUNG-LIE
WANG
T. POPOVICIU THEOREM [2, p. 1101. (a) Let 4 be a real-aalued function on [a, b] and x, , y, (j = I,..., n)2n points on [a, b] such that max{xl
thenifp,>O,j=l,...,
,.. ., 4
< minh
,...,m),
xl +
3-conerex
y1 = -** = x, + yn;
(3)
n,P,=xpp,,
If CJis strictly 3-convex, then equality occurs in (4) only af all x, are equal. (b) If for a continuous function 4 inequality (4) holds (strictly) for every n and all 2n (distinct) points satisfying (3) and all pi > 0, then 4 is (strictly) 3conzfex. An inequality converse to (4) can be found in Vasic and JaniC [7]. In view of the above, the Levinson inequality (2) which originated from the Ky Fan inequality (l), is well developed. On the other hand, the development of the remarkable Ky Fan inequality (1) came to a standstill after the elegant work of Levinson. It should be noted that, as indicated in [ 1, p. 51, inequality (1) was originally established by forward and backward induction as often used for establishing the A-G (abbreviation of arithmetic and geometric) inequality [I, p. 41. In fact, inequality (1) is a replica of the A-G inequality (see also below). It appears that the intention of Beckenbach and Bellman [I] is to display a strong resemblance between the Ky Fan inequality (1) and the A-G inequality. Since there are 12 proofs of the A-G inequality given in [I, Sects. 5161 alone, it is natural to ask the question: “Are there more proofs of inequality (1) in addition to the one by Levinson [3] and the original, unpublished one ?” In this note, we will give an affirmative answer to the above question. In Section 2, we will rewrite (I) in a little more general form (Eq. (5)) and call it a Ky Fan inequality of the complementary A-G type. We will also present two variants of (5) and an inequality of Rado type (see, e.g., [I, p. 12; 4, p. 61; 5, p. 941) related to its variant (Eq. (6)). In Section 3, we will conclude with a conjecture associated with this new development of (1) and some remarks.
2. MAIN
RESULTS
Let p =z (p, ,..., p,J and .x = (x1 ,..., x,) be two sequences of positive numbers such that s, E (0, 41, j = l,..., n. Let A, , Gk (resp. ,4; and Gk) designate
ON
A KY FAN
503
INEQUALITY
(resp. complementary) arithmetic and geometric means of the numbers x1 )..., xl: (resp. 1 - x1 ,..., 1 - xJ with unequal weights pi ,..., p, , i.e.,
-4, = A&;
p) = : p,s,/Pk )
G, = G,(x; p) = fi xf’+ 3=1
,=l
G; = G,( I - .v; p),
,-J;. = &( I - x; p), where P, = x:=, p, , k = 2 ,..., n. We now give a Ky Fan inequality
of complementary
A-G type as follows:
which is equivalent to (1) (if p, = 1/n for all j). Since A, + .4:, = 1, (5) is equivalent to
G, d AdG, + GJ
(6)
A;(G,
(7)
or
Moreover,
+ G)
< ‘X .
since G, < GL and A, < -4; , (6) and (7) yield
Inequalities (6) and (7) are two variants of (5). Because of their equivalence, it is necessary to prove only one of them. It is also very interesting and significant to note that, like the A-G inequality, the Ky Fan inequality of complementary A-G type possesses an inequality of Rado type (see, e.g., [I, p. 121) related to (6):
PnL%(G,+ ‘$3 - 61 3 f’,d&-dG,-1
+ G-d - Gn-,I.
(9)
(Note also that an inequality of Rado type related to (7) can be given likewise.) Naturally, the establishment of (6) follows inductively from (9). To establish (9) consider the function f defined by
f (.s) = 4(G, + G;) - G, . Since A, = (P,-l/P,)A,-l
_
GP,-IIP+,IP,. 11-I n
+ (p,/P,)xn
, G, = Gi:yJPnx$IPm, etc. We have
504
CHUNG-LIE WANG
Simple manipulations reveal that
and
where
and
Since it is easy to see that U, > 0 always (unless Xj = + , j = l,..., n), the minimum is attained for
at which Vn = 0, W,, > 0, d2f/dx,2 > 0, and so
f(Xn) >f(G) = +
n
(Gnel+
G~-l)-P,‘Pn[tln-l(G,-~
+ G-I) - G-,1- (10)
By use of the fact that G, + GI, < 1, k = l,..., n, (9) follows from (10) immediately.
3. REMARKS AND A CONJECTURE From the above results, it may be expected that (6) can be established by some other standard approaches used for establishing the A-G inequality (see, e.g. [l, 4, 51). However, it should be pointed out here that several unsuccessful attempts have been made to establish (6) by directly using some approaches given in [I, Sects. 5-161. For this reason, it appears necessary to modify somewhat the standard approaches used for establishing the A-G inequality in establishing (6).
ON A KY FAN INEQUALITY
505
Conjecture. Let C$be an increasing function satisfying all conditions assumed in Levinson [3] (see above). Then we have
Obviously, (6) is the case 4(u) = log u and b = 8 of (I 1). REFERENCES New 1. E. F. BECKENBACH AND R. BELLRIAN, “Inequalities, ” 2nd ed., Springer-Verlag, York, 1965. 2. P. S. BULLEN, An inequality of N. Levinson, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., Nos. 412-460 (1973), 109-112. 3. N. LEVINSON, Generalization of an inequality of Ky Fan, J. Math. Anal. Appl. 8 (1964), 133-134. 4. G. H. HARDY, J. E. LITTLEWOOD AND G. P~LYA, “Inequalities,” 2nd. ed., Cambridge Univ. Press, Cambridge, 1952. 5. D. S. MITRINOVIC, “Analytic Inequalities,” Springer-Verlag, Berlin, 1970. 6. T. POPO~ICIU, Sur une inegalitb de N. Levinson, Mathematics (CZuj) 6 (1964), 301-
306. 7. P. M. VA& AND R. R. JANIC, On an inequality Belgrade 10 (1970), 155-157.
of N. Levinson,
Publ. Inst. Math.