@article{Rashid2023, 
author = {Mohammad H. M. Rashid and Feras Bani-Ahmad},
title = {An estimate for the numerical radius of the Hilbert space operators and a numerical radius inequality},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {11},
pages = {26384-26405},
keywords = {numerical radius, convex function operator, mixed Schwarz inequality, Furuta inequality, Young inequality},
url = {https://www.sciopen.com/article/10.3934/math.20231347},
doi = {10.3934/math.20231347},
abstract = {We provide a number of sharp inequalities involving the usual operator norms of Hilbert space operators and powers of the numerical radii. Based on the traditional convexity inequalities for nonnegative real numbers and some generalize earlier numerical radius inequalities, operator. Precisely, we prove that if              A        i    ,            B        i    ,            X        i    ∈      B    (      H    ) (   i  =  1  ,  2  ,  ⋯  ,  n),    m  ∈      N  ,    p  ,  q  >  1 with        1    p    +      1    q    =  1 and    ϕ and    ψ are non-negative functions on    [  0  ,  ∞  ) which are continuous such that    ϕ  (  t  )  ψ  (  t  )  =  t for all    t  ∈  [  0  ,  ∞  ), then         w          2      r            (                  ∑                  i          =          1                          n                                      X                i                              A                i        m                              B                i              )    ≤            n              2        r        −        1              m        ∑          j      =      1              m            ‖                  ∑                  i          =          1                          n                            1        p                    S                  i          ,          j                          p          r                    +              1        q                    T                  i          ,          j                          q          r                      ‖    −      r    0        inf                  ‖                  ξ                ‖            =      1        ρ  (  ξ  )  ,where        r    0    =  min  {      1    p    ,      1    q    },        S          i      ,      j        =            X        i        ϕ    2        (                  |                                            A                        i                          j              ∗                                      |              )              X        i    ∗  ,        T          i      ,      j        =            (                                    A                    i                      m            −            j                                                B                    i                    )        ∗        ψ    2        (                  |                                            A                        i            j                          |              )              A        i          m      −      j                  B        i   and     ρ  (  ξ  )  =            n              2        r        −        1              m        ∑          j      =      1              m            ∑          i      =      1              n                  (                                    ⟨                                          S                                  i                  ,                  j                                r                            ξ              ,              ξ                        ⟩                                              p              2                                      −                              ⟨                                          T                                  i                  ,                  j                                r                            ξ              ,              ξ                        ⟩                                              q              2                                          )        2    .}
}