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We obtain inequalities of the form C |f (z) |ᵖ |dz| A (p) ₓ |f (z) |ᵖ |dz|, (p>1) where f is harmonic in the unit disk D, T is the unit circle, and C is any convex curve in D. Such inequalities were originally studied for analytic functions by R. M. Gabriel Proc. London Math. Soc. 28 (2), 1928. We show that these results, unlike in the case of analytic functions, cannot be true in general for 0< p 1. Therefore, we produce an inequality of a slightly different type, which deals with the case 0<p<1. An example is given to show that this result is "best possible", in the sense that an extension to p=1 fails. Then we consider the special case when C is a circle, and prove a refined result which surprisingly holds for p=1 as well. We conclude with a maximal theorem which has potential applications.
Suman Das (Tue,) studied this question.