Isoperimetric type inequalities for harmonic functions

For 01p>1, every function f∈hpf∈hp satisfies the inequality‖f‖b2p⩽ap‖f‖hp,‖f‖b2p⩽ap‖f‖hp, where ap>1ap>1 is a suitable constant depending only on p  . Furthermore, by using the Carleman inequality in the form ‖f‖b4⩽‖f‖h2‖f‖b4⩽‖f‖h2 with f∈H2f∈H2, we prove the following refinement of the above inequality for p=2p=2‖f‖b4⩽1.5+24‖f‖h2.