In the paper, two known Opial-type inequalities due to G.-S. Yang [Proc. Japan Acad. 42, 78-83 (1966; Zbl 0151.052)] and P. R. Beesack and K. M. Das [Pac. J. Math. 26, 215-232 (1968; Zbl 0162.079)] are generalized by replacing the involved power functions by some non-decreasing convex functions. The first theorem of the paper says essentially that:
If f is an absolutely continuous function on \([a,b]\) with \(f(a)=0\), and P is nondecreasing on [0,\(\infty)\), \(P(0)=0\), Q is decreasing and convex on [0,\(\infty)\) and \(Q(0)=0\), then \[ \int^{b}_{a}P(| f(x)|)Q(| f'(x)|)dx \]
\[ \leq (b-a)(P\circ (b-a)Q^{-1})_ 0^{(-1)}(\frac{1}{b-a}\int^{b}_{a}Q(| f'(x)|)dx) \]
\[ \leq \int^{b}_{a}(P\circ (b-a)Q^{-1})_ 0^{(-1)}\circ Q(| f'(x)|)dx, \] where \(Q^{-1}\) is the right-continuous inverse of Q and \[ (P\circ (b-a)Q^{-1})_ 0^{(-1)}(u):=\int^{u}_{0}P((b- a)Q^{-1}(t))dt. \] The above inequality reduces to the Yang’s inequality mentioned above when \(P(u)=u^ p,\quad Q(u)=u^ q,\quad p>0\) and \(q\geq 1.\) The second theorem states an extension of the known inequality obtained by Beesack and Das and which is too long to be cited here. Some Orlicz spaces with Billik norms and certain normalized complementary Young functions produced from P and Q are needed for the establishment of the second result.