In this paper, Sturmian comparison theory is developed for the pair of second order differential equations; first of which is the nonlinear differential equations of the form \begin{equation} (m(t)\Phi_\beta(y'))'+um^n_{i=1}q_i(t)\Phi_{lpha_i}(y)=0 \end{equation} and the second is the half-linear differential equations \begin{equation} (k(t)\Phi_\beta(x'))'+p(t)\Phi_\beta(x)=0 \end{equation} where $\Phi_*(s)=|s|^{*-1}s$ and $\alpha_1>\dots>\alpha_m>\beta>\alpha_{m+1}>\dots>\alpha_n>0$. Under the assumption that the solution of Eq. (2) has two consecutive zeros, we obtain Sturm-Picone type and Leighton type comparison theorems for Eq. (1) by employing the new nonlinear version of Picone’s formula that we derive. Wirtinger type inequalities and several oscillation criteria are also attained for Eq. (1). Examples are given to illustrate the relevance of the results.