Integrals of subharmonic functions along two curves (Q1319095)

The problem of the constant \(K\) determination which satisfies the inequality \[ \int_{C_ 0} u(z)| dz|\leq K \int_ C u(z)| dz| \] is considered in the paper. \(C_ 0\) is a circle, \(C\) a rectifiable curve surrounding \(C_ 0\), \(u(z)\) a complex function defined in \(\overline{D}\), the adherence of the interior \(D\) of \(C\). If \({\mathcal F}\) is the class of the functions \(u(z)\) defined in \(\overline{D}\), the author denotes with \(K({\mathcal F},C)\) the least constant satisfying the above inequality and he considers the unit circle \(| z|=1\) as \(C_ 0\). The main results of the paper are contained in four theorems with the following contents: (i) If \(C\) is convex and \({\mathcal F}\) is the class \(S^ +\) of the subharmonic positive functions in \(D\) and upper semicontinuous in \(\overline{D}\), \(K<2\); (ii) If \(C\) is a circle, then \(K({\mathcal A},C)\leq 1\), where \({\mathcal A}= \{v\mid v=e^ u\), \(u\in S\}\); \[ K(S^ +,C)^{-1}= \inf_{0\leq\theta\leq2\pi} |(\partial/\partial \theta)\phi(e^{i\theta})|, \tag{iii} \] where \(w=\phi(z)\) maps \(| w|<1\) conformally onto \(D\); (iv) \(K(S^ +,C)+ K({\mathcal H}^ +,C)\) and \(K({\mathcal A},C)= K({\mathcal P},C)\), where \({\mathcal H}^ +\) is the class of positive, harmonic functions in \(D\) and continuous in \(\overline{D}\), \({\mathcal P}= \{u\mid u(z)= | P(z)|\}\), \(P(z)\), a polynomial.