In this paper we obtain coefficient characterization‎, ‎extreme points and‎ ‎distortion bounds for the classes of harmonic $p-$valent functions‎ ‎defined by certain modified operator‎. ‎Some of our results improve‎ ‎and generalize previously known results‎.

We develop unified information-theoretic machinery for deriving lower bounds for passive and active learning schemes. Our bounds involve the so-called Alexander’s capacity function. The supremum of this function has been recently rediscovered by Hanneke in the context of active learning under the name of “disagreement coefficient.” For passive learning, our lower bounds match the upper bounds o...

Upper bounds on Newton’s, Poisson’s, and energetic coefficients of normal restitution for the frictional impact of rigid pendulum against a fixed surface are derived, demonstrating that the upper bound on Newton’s coefficient is smaller than 1, while the upper bound on Poisson’s coefficient is greater than 1. The upper bound on the energetic coefficient of restitution, which is a geometric mean...