Let \(\mathcal{U}\) be the class of all univalent functions in the unit disc \(\mathbb D\). If \(X\) is a space of analytic functions in the unit disc \(\mathbb D\) then an interesting problem consists in describing geometrically those \(f\in\mathcal{U}\cap X\). There is a good chance in terms of the maximum modulus \(M_\infty(r,f)\) or in terms of the length of the image of \(\{z\in\mathbb D: |z|=r\}\) by \(f\), that is, \(2\pi r M_1(r,f')\). Indeed, earlier results of G. H. Hardy and J. E. Littlewood [Math. Z. 34, 403–439 (1931; Zbl 0003.15601; JFM 57.0476.01)], H. Prawitz [Arkiv Mat. 20, A, Nr. 6, 12 S (1927; JFM 53.0307.02)] and C. Pommerenke [Math. Math. Ann. 145, 285–296 (1962; Zbl 0117.04302)] prove that if \(f\in\mathcal{U}\) then \[ \begin{split} \int_0^1 M_\infty^p(r,f)\,dr<\infty &\qquad\Leftrightarrow\qquad f\in H^p,\qquad 0<p<\infty, \\ \int_0^1 M_1^p(r,f')\,dr<\infty &\qquad\Leftrightarrow\qquad f\in H^p,\qquad 0<p<2. \end{split} \]
In this paper the authors obtain a good number of results of this type, one of them asserts that if \(f\in\mathcal{U}\), \(2\leq p<\infty\), and \(\alpha>-2\) then the following conditions are equivalent
(1)\(\int_{\mathbb D}|f'(z)|^p\,(1-|z|^2)^{p+\alpha}\,dA(z)<\infty,\)
(2)\(\int_0^1r(1-r^2)^{\alpha+1}\left(\int_{|z|<r}|f'(z)|^2|f(z)|^{p-2}\,dA(z) \right)\,dr<\infty,\)
(3)\(\int_0^1r(1-r^2)^{\alpha+1}\left(\int_{|z|<r}|f'(z)|^2\,dA(z) \right)^{p/2}\,dr<\infty,\)
(4) \(\int_0^1 M_\infty^p(r,f)(1-r^2)^{\alpha+1}\,dr<\infty,\)
which extends previous results obtained in [A. Baernstein II, D. Girela, J. Á. Peláez, Ill. J. Math. 48, No. 3, 837–859 (2004; Zbl 1063.30014)].
Among other nice theorems which can also be found in the paper, it is worth to be noticed the existence of a much smaller Möbius invariant subspace of the Bloch space than \(Q_p\), \(0<p<1\), still containing all univalent Bloch functions.