  Let $S^{(\Lambda)}$ denote the classical Littlewood-Paley square function
formed with respect to a lacunary sequence $\Lambda$ of positive integers.
Motivated by a remark of Pichorides, we obtain sharp asymptotic estimates of
the behaviour of the operator norm of $S^{(\Lambda)}$ from the analytic Hardy
space $H^p_A (\mathbb{T})$ to $L^p (\mathbb{T})$ and of the behaviour of the
$L^p (\mathbb{T}) \rightarrow L^p (\mathbb{T})$ operator norm of
$S^{(\Lambda)}$ ($1 < p < 2$) in terms of the ratio of the lacunary sequence
$\Lambda$. Namely, if $\rho_{\Lambda}$ denotes the ratio of $\Lambda$, then we
prove that $$ \sup_{\substack{ \| f \|_{L^p (\mathbb{T})} = 1 \\ f \in H^p_A
(\mathbb{T}) } } \big\| S^{(\Lambda)} (f) \big\|_{L^p (\mathbb{T})} \lesssim
\frac{1}{p-1} (\rho_{\Lambda} - 1 )^{-1/2} \quad (1<p<2)$$ and $$ \big\|
S^{(\Lambda)} \big\|_{L^p (\mathbb{T}) \rightarrow L^p (\mathbb{T})} \lesssim
\frac{1}{(p-1)^{3/2}} (\rho_{\Lambda} - 1 )^{-1/2} \quad (1<p<2)$$ and that the
exponents $r=1/2$ in $(\rho_{\Lambda} - 1 )^{-1/2} $ cannot be improved in
general. Variants in higher dimensions and in the Euclidean setting are also
obtained.
