A result about Young's inequality and several applications.

[section]1. Introduction

A series of the inequalities played an important role in various fields of mathematics. Among these we found the famous Young inequality

[lambda]a +(1 - [lambda])b [greater than or equal to] [a.sup.[lambda]] [b.sup.1-[lambda]], (1)

for nonnegative real numbers a, b and [lambda] [0,1].

The Young inequality was refined by F. Kittaneh and Y. Manasrah in [6], thus:

[lambda]a +(1 - [lambda])b [greater than or equal to] [a.sup.[lambda]] [b.sup.[1-[lambda]]] + r [([square root of a] - [square root of b]].sup.2] (2)

where r = min{[lambda], 1 - [lambda]}.

This inequality was generalized by S. Furuichi in [4], thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

for a1,...,[a.sub.n] [greater than equal to] 0 and [p.sub.1],...,[p.sub.n] [greater than or equal to] 0 with [p.sub.1] + ... + [p.sub.n] = 1, where [p.sub.min] = min{p1,...,[p.sub.n]}.

Another generalizations can be found by J. M. Aldaz in [1] and [2].

In [9], M. Tominaga, showed the reverse inequality for Young's inequality, using Specth's ratio, thus

S(a / b) [a.sup.[lambda]][b.sup.[1-[lambda]]] [greater than or equal to] [lambda]a + (1 - [lambda])b, (4)

where the Specht's ratio [8] was defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for a positive real number h.

S. Furuichi, in [5] given another type of the improvement of the classical Young inequality by Specht's ratios, thus

[lambda]a + (1 - [lambda])b [greater than or equal to] S ([(a / b).sup.r]) [a.sup.[lambda]] [b.sup.1-[lambda]] . (5)

In fact Young's inequality is a special case of the Jensen inequality. Therefore, we seek some improvements of this inequality in many papers and books.

A main result given by S. Dragomir [3], in general form, is studied by F. C. Mitroi [7] in a particular case, thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where f is a convex function, pi > 0 for all i = 1,...,n and [n summation over (i=1) pi = 1.

[section]2. Main results

Theorem 2.1. For a,b > 0 and [lambda] [member of] (0,1), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where r = min{[lambda], 1 - [lambda]}.

Proof. In inequality (6) for n = 2, [p.sub.1] = [lambda], [p.sub.2] = 1 - [lambda], with [lambda] [member of](0,1), [x.sub.1] = a, [x.sub.2] = b, f( x) = - log x and taking account that 1 - r = max{[lambda], 1 - [lambda]} when r = min{[lambda], 1 - [lambda]}, we deduce the inequality of the statement.

Remark 2.1.

a) Because a + b / 2 [greater than or equal to] [square root of ab], it follows that a + b / 2 [square root of ab] [greater than or equal to] > 1 and using inequality (7) we obtain the Young inequality.

b) In relation (7) we have equality if only if a = b.

Theorem 2.2. For x > -1 and [lambda] [member of] (0,1), we have the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where r = min{[lambda], 1 - [lambda]}.

Proof. If we take a / b = t in inequality (7), then we have the following inequality b

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

But, making the substitution t = x + 1 in relation (9) we have inequality (8).

Remark 2.2. Taking into account that [(x + 1).sup.2] + 1 / 2(x + 1) [greater than or equal to] 1, it is easy to see that inequality (8) is an improvement of the Bernoulli inequality (in the case [lambda] [member of](0, 1)). The equality holds when x = 1.

Theorem 2.3. Let p,q > 1 be real numbers satisfying 1 / p + 1 / q = 1. If [a.sub.i],[b.sub.i] > 0 for all

i = 1 , ..., n then there is the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. In Theorem 2.1 we take [lambda] = 1/p, which implies 1 - [lambda] = 1/q and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Making the sum for i = 1, ..., n we deduce inequality (10).

Remark 2.3.

a) It is easy to see that [greater than or equal to] > 1 and using inequality (10) we have a refinement of Holder's inequality.

b) In relation (10) the equality holds when a1 = ...= an and b1 = ...= [b.sub.n].

c) For p = q = 2 in inequality (10), we obtain a refinement of Cauchy's inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

Theorem 2.4. For any real numbers [a.sub.i], [b.sub.i]> 0, for all i = 1,...,n and p > 0, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. To prove this inequality, we will use the improvement of Holder's inequality from relation (10). We write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Right now we apply inequality (10), in the following way,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

But (p - 1)q = p, because 1/p + 1/q = 1. Adding relations (14) and (15), and taking into account that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we deduce the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

Dividing by [n.summation over (I=1)][([a.sub.i] + [b.sub.i]).sup.p] in relation (14), we obtain the inequality required.

Remark 2.4.

a) Since m [greater than or equal] 1, we have an improvement of Minkowski's inequality.

b) The equality holds in relation (13) for [a.sub.1] = ...= [a.sub.n] and [b.sub.1] = ...= [b.sub.n]. The integral versions of these inequality can be formulated as follows.

Theorem 2.5. Let p > 1 and 1 / p + 1 / q = 1. If f and g are real functions f,g [not equal to] 0 defined on

[a,b] such that [[absolute value of f].sup.p] and [[absolute value of g].sup.p] are integrable functions on [a, b], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

where r = min {1/p, 1/q},

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Equality holds iff [[absolute value of f(x)].sup.p] = [[absolute value of g(x)].sup.q]

Proof. We consider in Theorem 2.1 that [lambda] = 1/p and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By integrates from a to b in above inequality and by simple calculations, we deduce the inequality of statement. For [[absolute value of f(x)].sup.p] = [[absolute value of g(x)].sup.q] it is obvious that the equality holds.

Remark 2.5.

a) Because m [greater than or equal] 1 and according to inequality (17), we find a refinement for the integrated version of the Holder inequality.

b) For p = q = 2, we deduce a refinement for the integral version of the Cauchy inequality can be formulated as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Theorem 2.6. Let p > 1 and f, g [not equal to] 0, two real functions defined on [a, b] such that [[absolute value of f].sup.p] and [[absolute value of g].sup.p] are integrable functions on [a,b], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

where r = min{p, 1 - 1/p},

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. Since the Holder inequality is used to prove the Minkowski inequality, then we use Theorem 2.5, which is refinement of Hoolder's inequality, for to prove inequality (19). Therefore

[[absolute value of f(x) + g(x)].sup.p] [less than tha or equal to] [[absolute value of f(x)[parallel]f(x) + g(x)].sup.p-1] + [[absolute value of g(x)] x [[absolute value of f(x) + g(x)].sup.p-1],

it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We apply Theorem 2.5 in the following way:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In analogous way, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

But (p - 1)q = p. Therefore, adding inequalities (20) and (21), and taking into account that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] we deduce

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

Dividing the above inequality by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we obtain the inequality desired.

References

[1] J. M. Aldaz, Comparison of differences between arithmetic and geometric means, arXiv 1001. 5055v2, 2010.

[2] J. M. Aldaz, Self-improvement of the inequality between arithmetic and geometric means, Journal of Mathematical Inequalities, 3(2009), No. 2, 213-216.

[3] S. S. Dragomir, Bounds for the Normalized Jensen Functional, Bull. Austral. Math. Soc., 74(2006), No. 3, 471-478.

[4] S. Furuichi, A refinement of the arithmetic-geometric mean inequality, arXiv: 0912. 52 27v1, 2009.

[5] S. Furuichi, Refined Young inequalities with Specht's ratio, arXiv: 1004. 0581v2, 2010.

[6] F. Kittaneh and Y. Manasrah, Improved Young and Heinz inequalities for matrices, J. Math. Anal. Appl., 36(2010), 262-269.

[7] F. C. Mitroi, About the precision in Jensen-Steffensen inequality, Annals of the University of Craiova, Mathematics and Computer Science Series, 37(2010), No. 3, 73-84.

[8] W. Specht, Zer Theorie der elementaren Mittel, Math. Z., 74(1960), 91-98.

[9] M. Tominaga, Specht'ratio in the Young inequality, Sci. Math. Japon, 55(2002), 538- 588.

Nicusor Minculete

"Dimitrie Cantemir" University, Brasov 500068, Romania

E-mail: [e-mail]