Theorem 2.1 Let f and g be convex and increasing functions on [0, ∞) with f(0) = 0, and let p(s, t) ≥ 0, , D1D2p(s, t) > 0, s ∈ [α, a], t ∈ [β, b] with p(s, β) = p(α, t) = p(α, β) = 0 and D1D2p(s, t) |t = τ= 0. Further, let x(s, t) be absolutely continuous on [α, a) × [β, b], and x(s, β) = x(α, t) = x(α, β) = 0. Then, following inequality holds
(2.1)
Proof Let so that D1D2y(s, t) = |D1D2x(s, t)| and y(s, t) ≥ |x(s, t)|. Thus, from Jensen's integral inequality, we obtain
(2.2)
By using the inequality (2.2), we have
(2.3)
On the other hand
(2.4)
From (2.3) and (2.4), we have
This completes the proof.
Remark 2.2 Let x(s, t) reduce to s(t), and with suitable modifications in the proof of Theorem 2.1, then (2.1) becomes inequality (1.5) stated in Section 1.
Remark 2.3 Taking for g(x) = x in (2.1), then (2.1) becomes the following inequality.
(2.5)
Let x(s, t) reduce to s(t), and with suitable modifications, then (2.5) becomes inequality (1.4) stated in Section 1.
Remark 2.4 For f(t) = tl+1, l ≥ 0, the inequality (2.5) reduces to
(2.6)
In the right side of (2.6), by Hölder inequality with indices l + 1 and (l + 1)l, gives
(2.7)
Let x(s, t) reduce to s(t) and α = β = 0, then (2.7) becomes Hua's inequality (1.3) stated in Section 1.
Theorem 2.5
Assume that
-
(i)
f, g and x(s, t) are as in Theorem 2.1,
-
(ii)
p(s, t) is increasing on [0, a] × [0, b] with p(s, β) = p(α, t) = p(α, β) = 0,
-
(iii)
h is concave and increasing on [0, ∞),
-
(iv)
ϕ(t) is increasing on [0, a] with ϕ(0) = 0,
-
(v)
For ,
(2.8)
Then
(2.9)
where
(2.10)
(2.11)
and
Proof From (2.2), we easily obtain
(2.12)
From (2.8), (2.10-2.12) and Jensen's inequality(for concave function), hence
This completes the proof.
Remark 2.6 Let x(s, t) reduce to s(t), and with suitable modifications in the proof of Theorem 2.5, then (2.9) becomes the following inequality:
(2.13)
The inequality has been obtained by Rozanova in [17]. For and , the inequality (2.13) reduces to Polya's inequality (see [17]).
Remark 2.7 Taking for g(x) = x in (2.9), then (2.9) becomes the following interesting inequality.