Wiman’s type inequality in the unit ball and the diagonal maximal term - Basovskyi, Vitaliy; Bodnarchuk, Andriy; Skaskiv, Oleh; Trusevych, Oksana (PRECARPATHIAN BULLETIN OF THE SHEVCHENKO SCIENTIFIC SOCIETY. Number, 2026)
Let C^p be the p-dimensional complex vector space (p≥ 1), \|n\|=n_1+… +n_p, |z|=|z_1|^2+…+|z_n|^2 for n=(n_1,…,n_p)Z_+^p and z=(z_1,… , z_p)C^p, R_+= [0, +∞). In the paper we consider the class of analytic functions f, represented in the unit ball B_p={zC^p|z|<1} by power series of the form f(z)=_k=0^+∞ P_k(z); here P_0(z)a_0C, P_k(z) is a homogeneous polynomial of degree kZ_+. We denote M_f(r)={|f(z)||z|=r} and m(r,f)={|P_k(z)|k≥ 0}, the maximum modulus and maximal term of series, respectively; r[0, 1). In particular, the following statements are proved: 1^0. For every analytic function fA^p, p≥ 2 and for any ε>0 there exists a set E=E(ε, f)(0,1) of finite logarithmic measure such that the inequality M(r, f)≤ m(r, f)(1-r)^1+ε ((m(r, f)1-r))^1/2+ε holds for all r(0, 1)E. 2^0. Let h be a continuous positive increasing to +∞ on [0;1) functions such that ^1_0 h(r) d r =+∞. If the function fA^p is unbounded, then there exists a set E_1:=E(f,h)(0,1) such that M(r,f)≤ (1+o(1))(h(r)m(r,f)) (r→ 1-0, r(0,1)E_1), and h-meas E_1=_E_1(0,1)h(r)dr < +∞.