Strict inequalities for some critical exponents in two-dimensional percolation

For 2D percolation we slightly improve a result of Chayes and Chayes to the effect that the critical exponent for the percolation probability isstrictly less than 1. The same argument is applied to prove that ifL():={(x, y):x=r cos, y=r sin for some r0, or} and():=limpp _c log(p–p _c )]^–1 log P_cr {itO is connected to by an occupied path inL()}, then() is strictly decreasing in on 0, 2]. Similarly, lim_n –logn]^–1 logP _cr {itO is connected by an occupied path inL()() to the exterior of –n, n]×–n, n] is strictly decreasing in on 0, 2].