Further generalized refinement of Young’s inequalities for τ -mesurable operators

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.