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The authors find the greatest value  λ  and the least value  μ , such that the double inequalityC   ¯   (  λ  a  +  (  1  -  λ  b  )  ,  λ  b  +  (  1  -  λ  )  a  )  <  α  A  (  a  ,  b  )  +  (  1  -  α  )  T  (  a  ,  b  )  <    C   ¯   (  μ  a  +  (  1  -  μ  )  b  ,  μ  b  +  (  1  -  μ  )  a  )    holds for all  α  ∈ (0,1) and  a ,  b  > 0 with  a  ≠  b , whereC   ¯   (  a  ,  b  )  =  2  (    a    2    +  a  b  +    b    2    )  /  3  (  a  +  b  )   ,  A ( a ,  b ) = ( a  +  b )/2, and    T    a  ,  b    =2   /   π∫   0    π  /  2a    2cos    2    θ  +    b    2sin    2    θd  θ    denote, respectively, the centroidal, arithmetic, and Toader means of the two positive numbers  a  and  b .

Bounds for Combinations of Toader Mean and Arithmetic Mean in Terms of Centroidal Mean