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Proof (1) Using simple techniques of integration, we have the identity 1 d − c ∫ c d f ( g ( s ) ) d s = 1 d − c ∫ c c + d 2 [ f ( g ( s ) ) + g ( c + d − s ) ] d s. (2.3) .
Using simple techniques of integration, we have the identity 1 d − c ∫ c d f ( g ( s ) ) d s = 1 d − c ∫ c c + d 2 [ f ( g ( s ) ) + g ( c + d − s ) ] d s. (2.3).
Using simple techniques of integration and the hypothesis of p 1, we have the following identity: W P g ( t ) = ∫ c c + d 2 [ f ( t g ( c ) + ( 1 − t ) g ( s ) ) + f ( t g ( d ) + ( 1 − t ) g ( c + d − s ) ) ] p 1 ( s ) d s. for all t ∈ [ 0, 1 ].
Using simple techniques of integration and the hypothesis of p 1, we have the following identity: W H g ( t ) = ∫ c c + d 2 [ f ( t g ( s ) + ( 1 − t ) g ( c + d 2 ) ) + f ( t g ( c + d − s ) + ( 1 − t ) g ( c + d 2 ) ) ] p 1 ( s ) d s. for all t ∈ [ 0, 1 ].
Proof (1) Using simple techniques of integration and the hypothesis of p 1, we have the identities ∫ c d f ( g ( s ) ) p 1 ( s ) d s = ∫ c c + d 2 [ f ( g ( s ) ) + f ( g ( c + d − s ) ) ] p 1 ( s ) d s (3.3).
(2) Using simple techniques of integration, we have the following identity: P g ( t ) = 1 d − c ∫ c c + d 2 [ f ( t g ( c ) + ( 1 − t ) g ( s ) ) + f ( t g ( d ) + ( 1 − t ) g ( c + d − s ) ) ] d s .
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