Bir tüm fonksiyonun bir maksimum modül noktası ile sıfır kümesi arasındaki uzaklık

We obtain asymptotical bounds from below for the distance between a maximummodulus point and the zero set of an entire function. Known bounds (Macintyre,1938) are more precise, but they are valid only for some maximum moduluspoints. Our bounds are valid for all maximum modulus points and moreover, upto a constant factor, they are unimprovable.We consider entire functions of regular growth and obtain better bounds forthese functions. We separately study the functions which have very slow growth.We show that the growth of these functions can not be very regular and obtainprecise bounds for their growth irregularity.Our bounds are expressed in terms of some smooth majorants of the growthfunction. These majorants are defined by using orders, types, (strong) proximateorders of entire functions.Keywords: Entire function, Maximum modulus point, Zero set, Order, Type,Proximate order, Regular growth.iv