convexity, schur-convexity and bounds for the gamma ... - Project Euclid

1 downloads 0 Views 239KB Size Report
May 27, 1996 - functions. As a result, we re-derive, generalize or improve several inequalities due to Gautschi, Kershaw and Alzer. Also we present some new ...
               

     

                           ! " # $%  &   '&& (    ))  "   *%   *  & )$*  ()$*  &    ! " ()! '+  &%) )    ,! "  +  " % & "    *%  #* , 

 

  

   

      

               

                        

!!    "          # $          %  !& '

   ("          

)   " !  ! !  %  *   + ("           ("  ' ) #*    * -.! /001!   ) &  2# 34! /001 /00/   5&*! 33/6! )$*! -17-4 '(     ,&&  ! '&& &  )$*! 

 )$*!  " %% #*  8  #! ' &# 49 43! ' & ! '

&  &'! (%"( %$%"&) ()%"  !"#$% ­

,+

,-



             #       

 

   # . /     01         

-



 



  (   ' "    "                (   ' "      "     



    /          ' 0           /                    (   2 3  (         #        !              !       /     !  !      (          !   !       $%&' ( )* + ,% (  #    

         4    ! ,%%- 

,

    

    

   '   ' "   



                 

    '               

 

5   (            

2:; ( 2:; < 7 2>7

,,

     /  (! /     



        '   

 '  

        



               '   '    '         '      '      6 !                /                 #  

/     3              /           '                   ' 





7           -      ! #  ! /    (  !  ..(' 

     -             '

 

.     /      /    "    '  ' "    /     ! !  "   (   " / 8  9/  $   !    :     /  $                 /     !        







,9



:  !     !/                              - 2 ! /   (       '   !     !/                     ' 

   

     -                ( !!     5  ;  *      (       /            (  ..(' 



                                   



 











         '      

    ( '  "       

9



   



  "      "            ( '  "         "    "  

2:; ( 2:; < 7 2>7         

,
        +          

.    /         '    /                        

9



0     ;  *               -                    8      B       

      -/  $                                                                  /   !     !/           /                   2      /             Æ            #   /            !                ,     6     +      ! -                 !!    .        $( /       

   + #      /   !   !       /  !        !     

















.!    #  !             '      '               -' ' 

 9   -       !"





2:; ( 2:; < 7 2>7

9

.!  (              '     /            /           '      !   8  /  !     !                0        /       !     8 /        !    '/          9    (   ' "       "            '     2          '/

  !   !  !      #  ! +       #  ! '      !    / 

   9           - 0 !                (    /      !   !      



# ,$%&' ( +,% )* +  =          !  /  $    /        7    2 3  (    

 

 





 

 

 

 

    



       

          !     >  /     !C 1         7    

9'



2 3  (            7

9+

:  !     !/    " "   " /  !     2  "     /   "   "               #   /       '   ..(' #

          

 "    "  ' (       ( "  ' '   #       '     ! '           .!       '   '  2 3  (       /    

     '

,%%- ! 8!                         '         

+ 

''  

"    "  "       '

2  "        /  Æ    '+ "  '  "  "          .  $(    /  $    "     " '"     #    /               /     

9-



=    / ! #  ! -/              !!     . (! / !                '                              ' 

 

1 -% %2(.%* +         ( %2(.%* + .(3* '4%                        

,%%- 5

               

           

   $                               &  

',             



                  

'-





  : ;  *              /            

 /                              

                           









2:; ( 2:; < 7 2>7

9,

        ',   # 

     !  !           

  #  6/           ! '-  !   !      $(      2    !   !     3  (  1/  (! / #  ! 9            . (! /  #/             ( "  '

   3  (               /     #       /                      .    / #  ! 9              (  "      ' "    '    '  

              (  "    '

  "  '      ' 

         "    "               /           '       #     

   % 5  ?,9     !  /   !        

!!                  2!            !  #  !  . (! / !

99



;  *            /  $    '     '             ( ' "   '   "  '"      





        

DA.ADA68A2  #&"+    '!       !    ! '! /019  = +!      !  &%  /003 ! 33. 391  ? +   = &!      !  &%  /0@1 ! 160 11.   8A!  !      ! ?   %%  /0@- ! -6/ -6@  ,!              ! ?  5*  /060 ! .. @/  ? ,!       "  ! A A( A  /061 ! /./ /.-   = &!     !        "       ! ?   %%  /0@1 ! / 0

  '! #        !  &%  /0@9 ! 60. 144  7 "!  $  % " &       !  &%  /0@3 ! 14. 1//      2A! ' B (   )   ! & 5! "