Affiliation:
1. Department of Mathematics, Faculy of Sciences-Semlalia , University Cadi Ayyad , Av. Prince My. Abdellah, BP: 2390, Marrakesh (40.000-Marrakech) , Morocco .
Abstract
Abstract
In this paper, we prove that if a, b > 0 and 0 ≤ v ≤ 1. Then for all positive integer m
(1) - For v ∈
v
∈
[
0
,
1
2
n
]
v \in \left[ {0,{1 \over {{2^n}}}} \right]
, we have
(
a
v
b
1
-
v
)
m
+
∑
k
=
1
n
2
k
-
1
v
m
(
b
m
-
(
a
b
2
k
-
1
-
1
)
m
2
k
)
2
≤
(
v
a
+
(
1
-
v
)
b
)
m
.
{\left( {{a^v}{b^{1 - v}}} \right)^m} + \sum\limits_{k = 1}^n {{2^{k - 1}}{v^m}{{\left( {\sqrt {{b^m}} - \root {{2^k}} \of {\left( {a{b^{2k - 1}} - 1} \right)m} } \right)}^2} \le {{\left( {va + \left( {1 - v} \right)b} \right)}^m}.}
(2) - For v ∈
v
∈
[
2
n
-
1
2
n
,
1
]
v \in \left[ {{{{2^n} - 1} \over {{2^n}}},1} \right]
, we have
(
a
v
b
1
-
v
)
m
+
∑
k
=
1
n
2
k
-
1
(
1
-
v
)
m
(
a
m
-
(
b
a
2
k
-
1
-
1
)
m
2
k
)
2
≤
(
v
a
+
(
1
-
v
)
b
)
m
,
{\left( {{a^v}{b^{1 - v}}} \right)^m} + \sum\limits_{k = 1}^n {{2^{k - 1}}{{\left( {1 - v} \right)}^m}{{\left( {\sqrt {{a^m}} - \root {{2^k}} \of {\left( {b{a^{2k - 1}} - 1} \right)m} } \right)}^2} \le {{\left( {va + \left( {1 - v} \right)b} \right)}^m},}
we also prove two similar inequalities for the cases v ∈
v
∈
[
2
n
-
1
2
n
,
1
2
]
v \in \left[ {{{{2^n} - 1} \over {{2^n}}},{1 \over 2}} \right]
and v ∈
v
∈
[
1
2
,
2
n
+
1
2
n
]
v \in \left[ {{1 \over 2},{{{2^n} + 1} \over {{2^n}}}} \right]
. These inequalities provides a generalization of an important refinements of the Young inequality obtained in 2017 by S. Furuichi. As applications we shall give some refined Young type inequalities for the traces, determinants, and p-norms of positive τ-measurable operators.
Subject
Applied Mathematics,Control and Optimization,Numerical Analysis,Analysis
Reference12 articles.
1. [1] M. Akkouchi and M. A. Ighachane, A new proof of a refined Young inequality, Bull. Int. Math. Virtual Inst,. Vol. 10(3) (2020), 425-428.
2. [2] Y. Al- Manasrah and F. Kittaneh, A generalization of two refined Young inequalities, Positivity, 19(2015), 757-768.
3. [3] S. Furuichi, Alternative proofs of the generalized reverse Young inequalities, Adv. Inequal. Appl. 2017(2017)
4. [4] B. Fuglede, Rv. Kadison, On determinants and a property of the trace in finite factors, Proc Nat Acad Sci, 37(1951), 425-431.
5. [5] B. Fuglede, Rv. Kadison, Determinants theory in finite factors, Ann. Math, 55(1952), 520-530.
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