Some Hardy-type integral inequalities involving functions of two independent variables
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Positivity
Some Hardy-type integral inequalities involving functions of two independent variables Bouharket Benaissa1,2
· Mehmet Zeki Sarikaya3
Received: 24 May 2019 / Accepted: 10 October 2020 © Springer Nature Switzerland AG 2020
Abstract In this paper, we give some new generalizations to the Hardy-type integral inequalities for functions of two variables by using weighted mean operators S1 := S1w f and S2 := S2w f defined by 1 S1 (x, y) = W (x)W (y)
x x 2
y y 2
w(t)w(s) f (t, s)dsdt,
and S2 (x, y) =
x x 2
y y 2
w(t)w(s) f (t, s)dsdt, W (t)W (s)
with W (z) =
z
w(r )dr
f or z ∈ (0, +∞),
0
where w is a weight function. Keywords Hölder’s inequality · Fubini theorem · weight function Mathematics Subject Classification 26D15 · 26D10
B
Bouharket Benaissa [email protected] Mehmet Zeki Sarikaya [email protected]
1
Faculty of Material Sciences, University of Tiaret, Tiaret, Algeria
2
Laboratory of Informatics and Mathematics, University of Tiaret, Tiaret, Algeria
3
Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce, Turkey
123
B. Benaissa, M. Z. Sarikaya
1 Introduction The well-known generalized Hardy integral inequalities are as follows: Let f be a non-negative measurable function on (0, ∞), ([5]) Classical Hardy’s inequality Let ⎧ x ⎪ ⎪ f (t)dt f or m > 1, ⎨ 0 F(x) = ∞ ⎪ ⎪ ⎩ f (t)dt f or m < 1, x
then
∞
x −m F p (x)d x ≤
0
p |m − 1|
p
∞
x −m (x f (x)) p d x,
f or p > 1
(1)
0
Inequality (1) can be rewritten in the form Let ⎧ x ⎪ ⎪ f (t)dt f or α < p − 1, ⎨ 0 F(x) = ∞ ⎪ ⎪ ⎩ f (t)dt f or α > p − 1, x
then
∞
x α− p F p (x)d x ≤
0
p | p − 1 − α|
p
∞
x α f p (x)d x,
f or p > 1.
(2)
0
In 1995, Pachpatte proved the following theorem [11]. Let f be a nonnegative integrable function on = (0, a) × (0, b), where a, b are positive constants and r1 , r2 be positive and absolutely continuous functions on (0, ∞) such that
1+
1 py r2 (y) 1 px r1 (x) ≥ and 1 + ≥ , p − 1 r1 (x) α p − 1 r2 (y) β
for almost all x, y ∈ (0, ∞) and for some constants α > 0, β > 0, p > 1. If F is defined by 1 F(x, y) = r1 (x)r2 (y)
123
x x 2
y y 2
r1 (s)r2 (t) f (s, t) dtds, st
Some Hardy-type integral inequalities involving…
then
F(x, y) p d yd x xy 0 0 2 p a b p 1 p × |r2 (y)r1 (x) f (x, y) ≤ (αβ) p−1 x yr1 (x)r2 (y) 0 0
y x x y x x y p − r1 ( ) f −r2 ( ) f , y − r2 ( )r1 (x) f x, , d yd x. 2 2 2 2 2 2 2
a
b
For Hardy integral inequalities which lead to many papers dealing with the alternative proofs, various generalizations, and numerous variants and applications in analysis see, [1–4], [6–10], [12–15]. Convention 0 (i) We adopt the usual convention = 0. 0 (ii) The weight function is a non-negative continuous function. Throughout the paper, we will assume that the functions f are non-negative integrable on = (0, a) × (0, b).
2 Preliminaries In this section we give some lemmas which will be used frequently in the proof of the main theorems. Let 0 < a, b < ∞ and = (0, a) × (
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