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I am currently reading the 1998 article Dynamics of the Binary Euclidean Algorithm: Functional Analysis and Operators by Brigitte Vallée, which cites a 1928 article by R. M. Gabriel for the following inequality: if $f$ is a holomorphic function defined in an open ball $U$, $\Gamma$ is a circular contour contained in $U$, $\gamma$ is a convex contour enclosed by $\Gamma$, and $p>0$, then $$\int_\gamma |f(z)|^p|dz|\leq 2\int_\Gamma |f(z)|^p|dz|.$$ The text suggests that Vallée borrowed this reference from the 1969 PhD thesis of Howard J. Schwartz - which deals with composition operators on Hardy spaces - although I haven't seen this thesis myself. I found Gabriel's original paper somewhat dated in its terminology and presentation, and I am wondering whether a more up-to-date reference exists for this result, or whether it has been subsumed into a more general result which is now relatively well-known. Is anyone able to point me in the direction of a modern version of this inequality, or a textbook which contains this inequality? In Vallée's application and in the one which I am considering, it is sufficient to consider the case in which $\gamma$ is also circular.

Since the result holds for all $p>0$, I wonder whether the key property being used is subharmonicity rather than holomorphicity. In any case, I haven't been able to find this result either in books on Hardy spaces (the context in which it is applied by Vallée, and presumably also Schwartz) or in books on convex analysis; or if I have found it, its modern form is so far removed from Gabriel's statement that I was unable to recognise it. Anyway, I would be very grateful if someone could direct me to a more modern reference for this result.

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This has been called "Gabriel's problem" by Ana Granados, who has reviewed it in Michigan Math. J. 46, 461-487 (1999). I think you'll find all the background and recent developments, together with open problems, that you might want in her overview. There are several variations on this inequality, some for holomorphic and some for subharmonic functions. The inequality you give requires a holomorphic function and was proven by Gabriel in 1946.

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