# Turán theory

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

P. Turán introduced [a52] and developed (see [a9], [a10], [a11], [a12], [a13], [a14], [a16], [a21], [a22], [a23], [a24], [a25], [a26], [a27], [a28], [a29], [a30], [a31], [a32], [a33], [a34], [a35], [a36], [a37], [a38], [a39], [a40], [a41], [a46], and all papers by Turán mentioned below) the power sum method, by which one can investigate certain minimax problems described below. The method is used in many problems of analytic number theory, analysis and applied mathematics.

Let be a fixed set of integers. Let be fixed complex numbers and let be complex numbers from a prescribed set. Define the following norms:

Bohr norm: ;

minimum norm: ;

maximum norm: ;

Wiener norm: ;

separation norm: ;

Cauchy norm: ;

argument norm: . Turán's method deals with the following problems [a91].

1) Determine, for ,

 (a1)

where the infimum is taken over all complex numbers (two-sided direct problems).

2) Find the above minimum in (a1) over all complex numbers satisfying or ( "two-sided conditional problems" ).

3) For a given domain and , find

(one-sided conditional problems).

4) For a given weight function and , find

(weighted two-sided problems).

5) For a given domain and , find

(dual conditional problems).

6) Given polynomials and , , and , determine

and

(two-sided direct operator problems).

7) Given a domain and , find

and

where and are as above (one-sided conditional operator problems).

8) Given a finite set of integers, fixed complex numbers , , and two generalized power sums , , how large can the quantities

be made simultaneously depending only on , , , , and (simultaneous problems)?

9) Given two finite sets of integers and , fixed complex numbers , , , , and , what is

and what are the extremal systems (several variables problems)?

Turán and others obtained some lower bounds for some of the above problems.

Let be a pure power sum. Then

and

(see also [a4]). These results were obtained in the equivalent form with and , respectively.

Also, let , where . Then

 (a2)

F.V. Atkinson [a2] improved this by showing that . A. Biro [a3] proved that and that if is such that , , then

J. Anderson [a1] showed that if , then , and that if is a prime number, then this lies in ; he also proved that if , then there exists a such that

It is also known [a43] that, on the other hand, for infinitely many and that for large enough .

P. Erdös proved that

where is the solution of the equation , and L. Erdös [a15] proved that if is large enough, then , where is the solution of the equation .

E. Makai [a44] showed that

For generalized power sums , Turán proved that if , then

Makai [a45] and N.G. de Bruijn [a4] proved, independently, that can be replaced with , where . If, however, one replaces it with for any , then the above inequality fails. Turán also proved that if , then

G. Halasz showed that for any ,

S. Gonek [a18] proved that for all ,

In the case of the maximum norm, V. Sos and Turán [a46] obtained the following result. Let . Then for any integer ,

with . G. Kolesnik and E.G. Straus [a42] improved this by showing that one can take . On the other hand, Makai [a45] showed that for

the inequality fails for some and .

Considering different ranges for , Halasz [a19] proved that if , then

## Other norms and conditions.

The following results are obtained for two-sided problems with other norms and conditions.

A) ([a17], [a47], [a8], [a45]). Let be ordered so that . Assume that and . Then

B) ([a91]). Let be ordered as in A). Assume that and , let be the largest integer satisfying and let be the smallest integer satisfying (if such an integer does not exist, take ). Then

C) ([a12]). Let and let , , be such that

Then

D) ([a59]). If and , then

E) ([a8]). If and is such that , then there exists a such that

F) ((Halasz). Let and be non-negative integers, , and . Assume that . Then there exists an integer such that

G) (Turán). If , then the above inequality holds with instead of .

## Problems of type 3) and 7).

Assume that , , with , let be real numbers, and let for . Define for some fixed complex numbers . Assuming that , Turán proved that and , where

and the minimum is taken over all integers .

If , then the above inequalities hold with

Also, if are polynomials of degree , and , then and , where

and the range of is .

Assume now that . Let be as defined above, and assume for , where and . Assume also that , . Take any , satisfying and define , by

(If or do not exist, replace them with .) Put and

Then and . If , then the above result holds with .

J.D. Buchholtz [a5], [a6] proved that if , then

respectively, where the last result is the best possible.

R. Tijdeman [a47] proved the following result for "operator-type problems" .

Let be fixed complex polynomials of degree and let . Then for every integer , , and , the inequality

 (a3)

holds, where is defined above and the factor is the best possible; also, if , then (a3) holds with instead of .

J. Geysel [a17] improved the above constant to

Turán studied the other "operator-type problem" for . Let be fixed complex numbers and let be a polynomial with no zeros outside . Assume that , and . Then

 (a4)

with

In case of the maximum norm and , Turán proved (a4) with

He also proved the following "simultaneous problem" . Let . For any integers and there exist a such that the inequalities

and

hold simultaneously.