Let a function be regular in the disk |z| < 1. The radius of univalence 0.164 … of the family of f with |an| ≤ n (n ≥ 2) is, actually, the radius of star-likeness. The radius of univalence 1 - [k/(l+K)]½ of the family of f with |an| ≤ k (n ≥ 2), where K > 0 is a constant, is, actually, the radius of starlikeness. The radii of convexity of the two families are estimated from below.

For holomorphic functions f with {\rm Re}\{zf'(z)/f(z)\}>\alpha and {\rm Re}\{zf'(z)/f'(z)\} >\alpha-1 , (0\leq\alpha<1) , respectively, in \{|z|<1\} , estimates of \sup_{|z|<1}(1-|z|^{2})|f'(z)/f'(z)| are given. Functions Gelfer-close-t0-convex of exponential order (\alpha, \beta) will also be considered.

We perform the Hamiltonian analysis of unimodular gravity in terms of the connection representation. The unimodular condition is imposed straightforwardly into the action with a Lagrange multiplier. After classifying constraints into first-class and second-class, the canonical quantization is carried out. We consider the difference of the corresponding physical states between unimodular gravity and general relativity. *

Let S be the family of functions f(z) -z + ajz* + ... which are analytic and univalent in \z\ < 1. We find the valueas a function of r , 0 < r < 1. The known lower estimate of / " ? / _ l /<(z) l |dz| is improved. Relations with the growth theorem are considered and the radius of univalence of f(z)/z is discussed.For g analytic in D -{\z\ < 1}, we setWe call g Dirichlet-finite if A(l,g) < cx>. Let S be the family of functions for 0 < r < 1. For each r, 0 < r < 1, the maximum is attained only by the rotations of the Koebe function: Ke{z) -e~i e K{e ie z) , where 0 is real. We first prove:

A harmonic function u in R is said to be quasi-bounded ([13]) if it can be represented as: u = u x -u 2 , where Uj{j ~ 1,2) is the limiting function of a monotone non-decreasing sequence of non-negative and bounded harmonic functions in R.A closed polar set E in a Riemann surface R is a closed set in R such that for every open parameter disc V in R, there exists a superharmonic function s v > 0 defined in V with the property that s v = + °° at every point in V Π it, or equivalently, V Π £ is a set of capacity zero in V ([1], [2]). It is known that i? -E is connected. Tumarkin and Havinson [17] (resp. Parreau [13]) investigated the null set E in a plane domain (resp. in a Riemann surface) i? for the class 5 (resp. H p ) under the condition that E is a compact set of logarithmic capacity zero (resp. a closed, not necessarily compact, polar set) and proved: if an analytic function / defined in R -E belongs to the class S{R -E)