Sharp pointwise estimates for analytic functions by the L p -norm of the real part

We obtain sharp estimates of by the Lp -norm of on the circle , where , and α is a real valued function on DR . Here f is an analytic function in the disc whose real part is continuous on , ω is a real constant, and is orthogonal to some continuous function Φ on the circle . We derive two types of estimates with vanishing and nonvanishing mean value of Φ. The cases Φ = 0 and Φ = 1 are discussed in more detail. In particular, we give explicit formulas for sharp constants in inequalities for with p = 1, 2, ∞. We also obtain estimates for in the class of analytic functions with two-sided bounds of . As a corollary, we find a sharp constant in the upper estimate of by which generalizes the classical Carathéodory–Plemelj estimate with p=∞.