We prove some results for algebraic polynomials in the complex plane that
relate the L-norm of the polar derivative of a complex polynomial and the
polynomial under some conditions. The obtained results include several
interesting generalizations of some Zygmund-type integral inequalities for
polynomials and derive polar derivative analogues of some classical
Bernsteintype inequalities for the sup-norms on the unit disk as well.
An arc-representation of a graph is a function mapping each vertex in the graph to an arc on the unit circle in such a way that adjacent vertices are mapped to intersecting arcs. The width of such a representation is the maximum number of arcs passing through a single point. The arc-width of a graph is defined to be the minimum width over all of its arc-representations. We extend the work of Barát and Hajnal on this subject and develop a generalization we call restricted arc-width. Our main results revolve around using this to bound arc-width from below and to examine the effect of several graph operations on arc-width. In particular, we completely describe the effect of disjoint unions and wedge sums while providing tight bounds on the effect of cones.
In this paper, firstly we prove an integral identity that one can derive
several new equalities for special selections of n from this identity:
Secondly, we established more general integral inequalities for functions
whose second derivatives of absolute values are GA-convex functions based
on this equality.
AbstractIn this paper, we offer a new quantum integral identity, the result is then used to obtain some new estimates of Hermite–Hadamard inequalities for quantum integrals. The results presented in this paper are generalizations of the comparable results in the literature on Hermite–Hadamard inequalities. Several inequalities, such as the midpoint-like integral inequality, the Simpson-like integral inequality, the averaged midpoint–trapezoid-like integral inequality, and the trapezoid-like integral inequality, are obtained as special cases of our main results.
Generalizations of some Zygmund-type integral inequalities for polar derivatives of a complex polynomial