7/29/2019 Ozols Raitis

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On a Certain Inequality

Raitis Ozols, University of Latvia, rozolsq@inbox.lv

Elementary algebra forms the basis for mathematical analysis and calculus.

Inequalities are especially important here. A lot of them were invented and proved as

convenient tools for investigating the classical problems of continuous mathematics. On theother side, inequalities themselves can provide stimulus for serious research already on high-

school level. John Littlewood has said in his famous book [1] that some classical inequalities

still today could turn into new important theorems if given to great mathematician in the

moment of inspiration. So there is no wonder that most famous of them have a lot of proofs;

e.g., the inequality between arithmetic and geometric means has 50 of them (see [2]). The

Cauchy-Schwartz inequality is an another example (see [3]).

The methods of inequality proving are also of the main topics in the preparation of

students to mathematical competitions. There are many teaching aids developed for this. The

main methods usually considered are as follows:

a) identical transformations (often to the form 02 kx )b) strengthening of the inequality,c) mathematical induction,d) derivative and integral,e) interpretations,f) applications of other inequalities (Cauchy, Cauchy-Schwartz, Jensen, etc.).

Many books on special methods have been published, e.g., [4].

In this paper we consider a proof of a double inequality L(a, b) M0,5(a, b) P(a, b) for

positive a and b. Hereab

abbaL

lnln),(

= (ab) and L(a, a) = a is the logarithmic mean,

2

5,0

2

),(

+=

babaM is the square root mean and

=

ab

abbaP

/arctan4

),( (ab) and

P(a, a) = a is the Seiffert mean. The proof can be of interest because it starts from the obvious

inequality 22 )1(0)1(