Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Abstract   ,  consider the generalized Marcum  Q -function of order    ν  >  0    real, defined by     Q    ν    (  a  ,  b  )  =    1      a    ν  -  1       ∫   b    ∞      t    ν      e    -      t    2    +    a    2      2        I    ν  -  1    (  at  )  d  t  ,   where    a  ,  b  ⩾  0   ,      I    ν      stands for the modified Bessel function of the first kind and the right hand side of the above equation is replaced by its limiting value when    a  =  0   . Our aim is to prove that the function    ν      ↦        Q    ν    (  a  ,  b  )    is strictly increasing on    (  0  ,    ∞  )    for each    a  ⩾  0   ,    b  >  0   , and to deduce some interesting inequalities for the function      Q    ν     . Moreover,  present a somewhat new viewpoint of the generalized Marcum  Q -function, by showing that satisfies the new-is-better-than-used (nbu) property, which arises in economic theory.
['Yin Sun', 'Árpád Baricz']
I am writing a research paper on a idea Inequalities for the generalized Marcum Q-function
[{"from": "human", "value": "I am writing a research paper on a idea Inequalities for the generalized Marcum Q-function"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Abstract   ,  consider the generalized Marcum  Q -function of order    \u03bd  >  0    real, defined by     Q    \u03bd    (  a  ,  b  )  =    1      a    \u03bd  -  1       \u222b   b    \u221e      t    \u03bd      e    -      t    2    +    a    2      2        I    \u03bd  -  1    (  at  )  d  t  ,   where    a  ,  b  \u2a7e  0   ,      I    \u03bd      stands for the modified Bessel function of the first kind and the right hand side of the above equation is replaced by its limiting value when    a  =  0   . Our aim is to prove that the function    \u03bd      \u21a6        Q    \u03bd    (  a  ,  b  )    is strictly increasing on    (  0  ,    \u221e  )    for each    a  \u2a7e  0   ,    b  >  0   , and to deduce some interesting inequalities for the function      Q    \u03bd     . Moreover,  present a somewhat new viewpoint of the generalized Marcum  Q -function, by showing that satisfies the new-is-better-than-used (nbu) property, which arises in economic theory."}]