ON HOLOMORPHIC FUNCTIONS WITH CERTAIN EXTREMAL PROPERTIES OF ITS ABSOLUTE VALUES

This paper is concerned with a ’special class of holomorphic functions with extremal properties of its absolute values on arbitrary closed line segments in the complex plane. The main result is a geoaz+b n metrical characterization of the functions z e z (az+b) and z (az+b) +i8 with a,b ,8 E IR n E Z

ABSTRACT.This paper is concerned with a 'special class of holomorphic functions with extremal properties of its absolute values on arbitrary closed line segments in the complex plane.The main result is a geo-az+b n metrical characterization of the functions z e z (az+b) and z (az+b) +i8 with a,b ,8 E IR n E Z KEY WORDS AND PHRASES.Maximum respectively minimum of the absolute value fl is taken on at one of the endpoints of every closed line segment.
1880 MATHEMATICS SUBJECT CLASSIFICATION CODE.

30C45 INTRODUCTI ON
The present work is closely related to the following problem raised by Rubel [I Find all entire functions f such that for every closed line segment L in the complex plane, wherever located an in whatever direction, the maximum of fl on L is taken on at one of the two endpoints of L As a secondary result of the solution for this problem we will ob- az+b tain a simple characterization of the entire function z e Suppose f G is a complex function holomorphic in the region G Then for all z=x+iyG with f(z) # O the partial derivatives of first and second order of the function w(x,y) If(z) are given by [2]: Moreover the formula of Taylor implies: w(x+h,y+k) w(x,y) + hw x(x,y) + kWy(X,y) + I {h2Wxx(X,y)+ 2hkWxy(X,y)+ k2Wyy(X,y)} Introducing the variable := h+ik we deduce from (I.I) and (1.2) By means of this equation we prove the following Lemma.

LEMMA I. Let
Then there exists a line segment L through z such that fl does not reach its maximum respectively minimum at one of the endpoints of L PROOF.Suppose z G with f(z) $ 0 f' (z) $ 0 and Re(f (,z) f" (z)) > f'2(z) For real t in a sufficiently small neighbourhood of zero, we define fz) t Then we have: Re[{(z) ) 0 Is(f'f(z)(z) [, t   f"() 2) t 2 f(z.)f_"(z).) Re[f(Z Re(f'2(z) From these equations it follows by means of (I .3) Hence there exists a t o R such sthat: The case Re[ f (z)f''(z)) < is treated in the same way.
'f, 2 (z) The next Lemma is an immediate consequence of the well known theorem of Picard [3], "Let g be a meromorphic function in the whole complex plane.If there exist three different numbers not belonging to the range of g then g is constant".
LEMMA 2. Let f be a meromorphic function in the whole complex plane, which is not constant.f f,, Then the function g := ---, is either a constant or there exist z O, z { with Re(g(z O)) > and Re(g(zl < PROOF.Since f is meromorphic in all of and not constant, f f,, also the function g -is meromorphic in all of f, Then our Lemma immediately follows from the theorem of Picard.
Collecting the results obtained so far we end up with the following theorem: THEOREM I. Suppose f is a non constant function meromorphic in ff,, the whole complex plane such that also g is not a constant.
Then there exist two lime segments L and L such that neither the maximum of f[ on L O nor the minimum of f on L is taken on at the endpoints of these segments.
f f" Next we consider the case that the expression is a constant on -r c - (1.7)   it may easily be integrated [4].
The result is (1.6) with a 6 {o} and b E In the case c y+i the introduction of new variables u and 8 by u + i8 := I---leads to the relations and thereby (1.8) (,=Oy=l c,>Owy< e<Ol <y (I .9) which will be needed later on.z The investigation of the functions f(z) e respectively f(z) z with regard to the extremal properties of their absolute values causes no difficulties.Since the simple similarity transformation z az+b maps line segments into line segments there directly follows: THEOREM 2. If f is a non-constant, entire function on such that on every line segment L its absolute value ]fl takes on its maximum at one of the endpoints of L then f is given either by az+b n f(z) e or by f(z) (az+b) n l Theorem 2 completely solves the problem of Rubel mentioned at the be- ginning.In view of the equation f(z) f(z)------T a further con- sequence of Theorem 2 is: THEOREM 3. If f is a non constant function meromorphic in the entire complex plane such that on every line segment L its absolute value fl reaches its minimum at one of the endpoints of L then f is given either by f(z) e az+b or by f(z) n n 6 lq (az+b) The combination of theorem 2 and Theorem 3 leads to a simple charac- terization of the exponential function: THEOREM 4. Let f be a non-constant entire function such that on every line segment L the absolute value fl reaches its maximum as well as its minimum at the endpoints of L then f is an expo-az+b nential function of the form f(z) e In view of Lemma 3 it seems to be interesting to investigate the gen- On the half-lines with const, the behaviour of fl is obvious.
In the case of straight lines not running through the origin we have to consider separately those cutting the negative real axis.Finally in view of (1.11) it suffices to investigate If[ on straight lines cutting the positive real axis vertically respectively on half-lines cutting the negative real axis vertically.
A straight line of the first kind is given in polar coordinate by: r __2 cos p > O < < (1.12) For e > O it follows by means of elementary analysis that there exists exactly one minimum of f on (I 12) given by: tan e (1.13) Similarly for e < O there exists exactly one maximum of fl on (1.12) also fixed by (1.13).These results are in accordance with Lemma and equation (I .9).Moreover the result (1.13) may be easily derived also via geometrical arguments by considering the geometry of the set of curves r e e -8 const.THEOREM 5. Let G be the region defined by (I. 10),K the class of all functions f holomorphic and non-constant in G with the ff,.
further property that g ,2 is meromorphic in the entire complex plane.Every function f K such that on any line segment L c G its absolute value fl reaches its maximum (respectively minimum)   in one of the endpoints of L is given by f(z) e az+b or f(z)=(az+b) e+i8 with e > O (respectively f(z) e az+b or f(z)=(az+b)e+i8 with e <0) In passing it should be mentioned that.Ullrich [4 in his paper "Betragflchen mit ausgezeichnetem KrHmmungsverhalten" ends up with the same functions which I have discussed in my paper [5], too.

Mathematical Problems in Engineering
Special Issue on Time-Dependent Billiards

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: 1-c for c # PROOF.Rewriting the differential equation (1.5) in the form f,, f, For > e,,e O there occur two turning points, the position of which is fixed by: the half-lines mentioned above the arguments have to be slightly modified because of the limits: Departamento de Estatística, Matemática Aplicada e Computação, Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, Avenida 24A, 1515 Bela Vista, 13506-700 Rio Claro, SP, Brazil ; edleonel@rc.unesp.brAlexander Loskutov, Physics Faculty, Moscow State University, Vorob'evy Gory, Moscow 119992, Russia; loskutov@chaos.phys.msu.ru