times Differentiable - Convex functions in the first sense

Turkish Journal of Analysis and Number Theory, 2017, Vol. 5, No. 2, 63-68 Available online at http://pubs.sciepub.com/tjant/5/2/4 ©Science and Education Publishing DOI:10.12691/tjant-5-2-4

Some New Integral Inequalities for 𝒏𝒏-times Differentiable 𝒔𝒔- Convex functions in the first sense Mahir Kadakal1,*, Huriye Kadakal2, İmdat İşcan1

1

Department of Mathematics, Faculty of Sciences and Arts, Giresun University-Giresun-TÜRKİYE 2 Institute of Science, Ordu University-Ordu-TÜRKİYE *Corresponding author: [email protected]

Received January 03, 2017; Revised February 04, 2017; Accepted February 13, 2017

Abstract In this work, by using an integral identity together with both the Hölder and the Power-Mean integral inequality we establish several new inequalities for n-time differentiable 𝑠𝑠-convex functions in the first sense.

Keywords: convex function, 𝑠𝑠- convex function in the first sense, hölder integral inequality and power-mean integral inequality

Cite This Article: Mahir Kadakal, Huriye Kadakal, and İmdat İşcan, “Some New Integral Inequalities for 𝑛𝑛times Differentiable 𝑠𝑠- Convex functions in the first sense.” Turkish Journal of Analysis and Number Theory, vol. 5, no. 2 (2017): 63-68. doi: 10.12691/tjant-5-2-4.

1. Introduction In this paper, by using the some classical integral inequalities, Hölder and Power-Mean integral inequality, we establish some new inequalities for functions whose 𝑛𝑛th derivatives in absolute value are 𝑠𝑠-convex functions in the first sense. For some inequalities, generalizations and applications concerning convexity see [1-11]. Recently, in the literature there are so many papers about 𝑛𝑛 -times differentiable functions on several kinds of convexities and 𝑠𝑠 -convex functions. In references [5,6,7,8], readers can find some results about this issue. Many papers have been written by a number of mathematicians concerning inequalities for different classes of convex and 𝑠𝑠-convex functions in the first sense see for instance the recent papers [12-19] and the references within these papers. There are quite substantial literatures on such problems. Here we mention the results of [1-19] and the corresponding references cited therein. Definition 1.1: A function 𝑓𝑓: 𝐼𝐼 ⊆ ℝ → ℝ is said to be convex if the inequality

f ( tx + (1 − t ) y ) ≤ tf ( x ) + (1 − t ) f ( y )

is valid for all 𝑥𝑥, 𝑦𝑦 ∈ 𝐼𝐼 and 𝑡𝑡 ∈ [0,1] . If this inequality reverses, then 𝑓𝑓 is said to be concave on interval 𝐼𝐼 ≠ ∅. This definition is well known in the literature. Definition 1.2: A function 𝑓𝑓: [0, ∞) → ℝ is said to be 𝑠𝑠-convex in the first sense if f (α x + β y ) ≤ α s f ( x ) + β s f ( y ) 𝑠𝑠

𝑠𝑠

holds for all 𝑥𝑥, 𝑦𝑦 ∈ [0, ∞), 𝛼𝛼, 𝛽𝛽 ≥ 0 with 𝛼𝛼 + 𝛽𝛽 = 1 and for some fixed 𝑠𝑠 ∈ (0,1]. It can be easily seen that every 1-convex function is convex.

In [21], Dragomir and Fitzpatrick proved a variant of Hadamard’s inequality which holds for 𝑠𝑠-convex mapping in the first sense. Definition 1.3: The following double inequality is wellknown in the literature as Hadamard’s inequality for convex mappings [8,10]: Let 𝑓𝑓: 𝐼𝐼 ⊆ ℝ → ℝ be a convex mapping defined on the interval 𝐼𝐼 in ℝ and 𝑎𝑎, 𝑏𝑏 ∈ 𝐼𝐼 with 𝑎𝑎 < 𝑏𝑏. Then the following inequality holds: b f ( a ) + sf ( b ) 1  a+b f ≤ f ( x ) dx ≤ .  ∫ − b a s +1 2   a

Throughout this paper we will use the following notations and conventions. Let 𝐽𝐽 = [0, ∞) ⊂ ℝ = (−∞, +∞) , and 𝑎𝑎, 𝑏𝑏 ∈ 𝐽𝐽 with 0 < 𝑎𝑎 < 𝑏𝑏 and 𝑓𝑓′ ∈ 𝐿𝐿[𝑎𝑎, 𝑏𝑏] and 1

 b p +1 − a p +1  p a+b A ( a, b ) = = , L p ( a, b )   ,  ( p + 1)(b − a )  2   a ≠ b, p ∈ , p ≠ −1, 0

be the arithmetic and generalized logarithmic mean for 𝑎𝑎, 𝑏𝑏 > 0 respectively. We will use the following Lemma [20] for we obtain the main results: Lemma 1.1: Let 𝑓𝑓: 𝐼𝐼 ⊆ ℝ → ℝ be 𝑛𝑛-times differentiable mapping on 𝐼𝐼 ° for 𝑛𝑛 ∈ ℕ and 𝑓𝑓 (𝑛𝑛) ∈ 𝐿𝐿[𝑎𝑎, 𝑏𝑏] , where 𝑎𝑎, 𝑏𝑏 ∈ 𝐼𝐼 ° with 𝑎𝑎 < 𝑏𝑏, we have the identity n −1

 f ( k ) ( b ) b k +1 − f ( k ) ( a ) a k +1  b  − f ( x )dx   ∫a k + 1) ! (  

k ∑ ( −1) 

k =0

=

( −1)n+1 n!

b n

∫a x

n f ( ) ( x )dx

where an empty sum is understood to be nil.

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1

2. Main Results Theorem 2.1. For ∀𝑛𝑛 ∈ ℕ ; let 𝑓𝑓: 𝐼𝐼 ⊂ [0, ∞) → ℝ be 𝑛𝑛 -times differentiable function on 𝐼𝐼 ° and 𝑎𝑎, 𝑏𝑏 ∈ 𝐼𝐼 ° with 𝑞𝑞 𝑎𝑎 < 𝑏𝑏 . If �𝑓𝑓 (𝑛𝑛) � for 𝑞𝑞 > 1 is 𝑠𝑠 -convex function in the first sense on [𝑎𝑎, 𝑏𝑏], then the following inequality holds:

 ( k ) ( b ) b k +1 − f ( k ) ( a ) a k +1  b k f  − f ( x)dx ∑ ( −1)   ∫ k + 1 ! ( ) k =0   a n −1

b p  1  x np +1  = n !  np + 1  a 

q  (n)  f ( b ) ( x − a ) s +1  s +1  (b − a ) s  

q

n f ( ) (a)  ( x − a ) s +1  + (b − a ) s x −   s + 1  (b − a ) s 

1

q q q  (n) (n) 1  f (b ) + s f ( a )  1  , ≤ ( b − a ) q Lnnp (a, b)  n! s +1     where 1/ p + 1/ q = 1. 𝑞𝑞

Proof. If �𝑓𝑓 (𝑛𝑛) � for 𝑞𝑞 > 1 is 𝑠𝑠-convex function in the first sense on [𝑎𝑎, 𝑏𝑏] , using Lemma1.1, the Hölder integral inequality and

b−x  n  x−a f( ) b+ a b − a b −a  

q

n = f ( ) ( x)

 f ( k ) ( b ) b k +1 − f ( k ) ( a ) a k +1  b  − f ( x)dx   ∫ k + 1)! (   a

k ∑ ( −1) 

k =0

1 (b − a ) n!

1 p

1

1

1

1

1

1 1 (b − a ) q n!

n −1

k =0

1

q  (n)  f (b ) b  ( x − a )s dx ∫ s  (b − a ) a   1

q n f ( ) (a) b  (b − a ) s − ( x − a ) s  dx  + ∫   (b − a ) s a    q

 f ( k ) ( b ) b k +1 − f ( k ) ( a ) a k +1  b  − f ( x)dx   ∫ k + 1) ! (   a

1 q q  n 1 n ≤ ( b − a ) q Lnnp (a, b) A q  f ( ) ( a ) , f ( ) ( b )  n!  

  x − a  q  (n)      f (b ) b   b − a    × ∫   dx  s q  a    x − a   (n)     + 1 −  b − a   f ( a )             1

q q q  (n) (n)  f (b) + s f ( a )   . Lnnp (a, b)  s +1    

k ∑ ( −1) 

1 q

b 1  np  p  ∫x dx   n!  a 

 b np +1 − a np +1  p    (np + 1)(b − a ) 

This completes the proof of theorem. Corollary 2.1. Under the conditions of Theorem 2.1 for 𝑠𝑠 = 1, we obtain the following inequality

q b n q  ∫ f ( ) ( x ) dx    a 

b p 1 ≤  ∫x np dx   n!  a  s

q  (n)  f ( b ) (b − a ) s +1  s s +1  (b − a) 

q q q  n n f ( ) (b) + s f ( ) ( a )    × ( b − a ) s +1    

a

b p 1 ≤  ∫x np dx   n!  a 

     a

1

=

b

1 n ≤ ∫x n f ( ) ( x ) dx n!

1 b q

1

=

we have

a

q q n s +1 f ( ) (a)    b − a) ( s s (b − a) b − + − (b − a) a   s +1  (b − a) s   

q

s s q  q  x − a  (n)  x − a   (n) f b ≤ +  1 − ( )     f (a) , b−a   b − a  

n −1

1

1  b np +1 − a np +1  p =    n !  np + 1 

b

which coincide with the Theorem 2.1 in [20]. Proposition 2.1. Under the conditions of Corollary 2.1 for 𝑛𝑛 = 1, we obtain the following:

f ( b ) b − f (a)a b−a 1

≤ (b − a ) q

−1

b

−

1 f ( x)dx b−a ∫ a

1 q

q q  n n L p ( a, b) A  f ( ) ( a ) , f ( ) ( b )  .  

Proposition 2.2. For 𝑞𝑞 = 1 , we obtain the following inequality:

Turkish Journal of Analysis and Number Theory

f ( b ) b − f (a)a

65

b

1

1 f ( x)dx b−a ∫

q   ( n) q f (b ) b−a     a   b s   (b − a )   s ≤ L p ( a, b ) A f ' ( a ) , f ' ( b ) . 1 x − a ) x n dx  (    1 − ∫ b q  1  n  q  f ( n) ( a )  a  ∫x dx  ×    be Theorem 2.2. For ∀𝑛𝑛 ∈ ℕ ; let f : I ⊂ [ 0, ∞ ) →=   n!  −   a   s   (b − a )   n -times differentiable function on I  and a, b ∈ I  with   (𝑛𝑛) 𝑞𝑞 (𝑛𝑛) 𝑞𝑞 b q a < b . If �𝑓𝑓 � ∈ 𝐿𝐿[𝑎𝑎, 𝑏𝑏] and �𝑓𝑓 � for 𝑞𝑞 ≥ 1 is   n) ( n 𝑠𝑠-convex function in the first sense on [𝑎𝑎, 𝑏𝑏], then the  + f ( a ) ∫x  a   following inequality holds:

−

)

(

1

 ( k ) ( b ) b k +1 − f ( k ) ( a ) a k +1  b k f  − f ( x)dx ∑ ( −1)   ∫ 1 ! k + ( ) k =0   a

q   ( n) q  f (b )      s   (b − a )   1  K ( s , n, x )    1 − q b q   n  1  x n +1   f ( ) ( a )    = ×   −   n!  n + 1   s    a (b − a )        b  1 n + q x   n) (   + f ( a )  n + 1       a  

n −1

q −1

n 1 ≤ (b − a ) Ln q (a, b) n! 1

q q  ( n) q (n)  f (b ) − f ( a )     K ( s , n, x )     (b − a ) s +1 ×    .     q   n + f ( ) ( a ) Lnn (a, b)   

p= 1−

Where

1 q

𝑏𝑏

and

𝑝𝑝 > 1

,

1−

𝐾𝐾(𝑠𝑠, 𝑛𝑛, 𝑥𝑥) =

∫𝑎𝑎 (𝑥𝑥 − 𝑎𝑎) 𝑠𝑠 𝑥𝑥 𝑛𝑛 𝑑𝑑𝑑𝑑. Proof. From Lemma1.1 and Power-mean integral inequality, we obtain

 f ( k ) ( b ) b k +1 − f ( k ) ( a ) a k +1  b  − f ( x)dx   ∫ k + 1) ! (   a

n −1

k ∑ ( −1) 

k =0

a

1

q  n n f ( ) (b ) f ( ) (a)    K ( s , n, x )  − (b − a ) s (b − a ) s      q  b n +1 − a n +1   n) ( f (a)      1 + n    q

q

1−

1

1

1

q q  ( n) q n f ( ) (a)   f (b )    K ( s , n, x )  − s +1 (b − a ) s +1    (b − a )  ×       q  b n +1 − a n +1    n) (  + f ( a )    (n + 1)(b − a )   

b  q 1 ≤  ∫x n dx   n!  a 

1

 q  x − a  s ( n )  q  f b    ( )   b n  b − a    ×  ∫x   dx  s q  a    x − a   (n)     + 1 −  b − a   f ( a )         

q −1

n 1 (b − a ) Ln q (a, b) = n!

q 1 1−  ( n ) b f (b ) b q    1  =  ∫x n dx  ( x − a )s x n dx  ∫ s   n!  (b − a ) a a  

1

1

q n f ( ) (a) b  s (b − a ) s − ( x − a )  x n dx  + ∫  (b − a ) s a    q

     ×     + 

 b n +1 − a n +1  q 1 (b − a )  =  n!  (n + 1)(b − a ) 

b

1 n ≤ ∫x n f ( ) ( x ) dx n! 1−

1

1  b n +1 − a n +1  q =    n !  n +1 

q q  ( n) q (n)  f (b ) − f ( a )     K ( s , n, x )     (b − a ) s +1 ×        q   n) ( n + f ( a ) Ln (a, b)   

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Turkish Journal of Analysis and Number Theory

This completes the proof of theorem. Corollary 2.2. Under the conditions of Theorem 2.2 for 𝑠𝑠 = 1, we obtain the inequality

 f ( k ) ( b ) b k +1 − f ( k ) ( a ) a k +1  b   − f ( x)dx 1 − ( ) ∑   ∫ 1 ! + k ( ) k =0   a n −1

≤

k

n 1 ( b − a ) Ln n!

q −1 q

( −1)n+1 b n!

∫x

n

n f ( ) ( x ) dx

a 1

b p 1 ≤  ∫1 p dx   n!  a 

1

( a, b )

1

q q  ( n ) (n)  f ( b ) − f ( a )  b−a  ×    q  (n) n  + f ( a ) Ln ( a, b ) 

   Lnn++11 ( a, b ) − aLnn ( a, b )   

(

1 q

)

    .    

b p 1 ≤  ∫1.dx   n!  a 

1

b p 1 =  ∫1.dx   n!  a 

Proposition 2.3. Under the conditions of Corollary 2.2 for 𝑛𝑛 = 1, we obtain the inequality

f ( b ) b − f (a)a b−a

q q  (n) (n)  f (b ) − f ( a ) b s  x nq ( x − a ) dx ∫ s  (b − a ) a   n + f ( ) (a)

b

1 f ( x)dx − b−a ∫

q nq ∫x dx  a 

qb

1

1 q

q  1 ' 1 (2b + a ) f ( b )   1  q 1− q ≤   A ( a, b)   . 6  +(2a + b) f ' ( a ) q   

Proposition 2.4. For 𝑞𝑞 = 1 , we obtain the following inequality f ( b ) b − f (a)a

 q  x − a  s ( n )  q     f b ( )   b nq  b − a     ∫x   dx  s q a    x − a   (n)    + 1 −  b − a   f ( a )         

1

a

b−a

1

q b q  ∫x nq f ( n ) ( x ) dx    a 

b

1 f ( x)dx − b−a ∫ a

≤ (2b + a ) f ( b ) + (2a + b) f ' ( a ) .

=

1 1 (b − a ) p n!

=

1 1 (b − a ) p n!

q q  (n) q (n)  f (b ) − f ( a )   S ( n, s , q , x )  s   (b − a )   + + nq nq 1 1 q b  −a   n) (   + f ( a )    nq + 1     q q  (n) (n)  f (b ) − f ( a )  S ( n, s , q , x )  (b − a ) s 

'

Theorem 2.3. For ∀𝑛𝑛 ∈ ℕ ; let f : I ⊂ [ 0, ∞ ) →  be

n -times differentiable function on I  and a, b ∈ I  with

1

n + (b − a ) f ( ) ( a )

q

 b nq +1 − a nq +1   q    (nq + 1)(b − a )  

1

q q  (n) q (n) a < b . If �𝑓𝑓 � ∈ 𝐿𝐿[𝑎𝑎, 𝑏𝑏] and �𝑓𝑓 � for 𝑞𝑞 > 1 is  f (b ) − f ( a )   S ( n, s, q, x )  𝑠𝑠-convex function in the first sense on [𝑎𝑎, 𝑏𝑏], then the 1 + s 1 =  (b − a ) (b − a )  following inequality holds: n!   q  + f ( n ) a Ln  n −1  f ( k ) ( b ) b k +1 − f ( k ) ( a ) a k +1  b ( ) nq k     − ∫ f ( x)dx ∑ ( −1)   + k 1 ! ( ) k =0 Corollary 2.3. Under the conditions of Theorem 2.3 for   a 𝑠𝑠 = 1 we obtain the inequality 1 q q q  (n)  (n)  f (b ) − f ( a )  n −1  ( k ) ( b ) b k +1 − f ( k ) ( a ) a k +1  b k f  − f ( x)dx   S n s q x , , , ) ( 1 ∑ ( −1)  ≤ (b − a )   ∫  1 ! k + (b − a ) s +1 ( ) k =0   a n!   q  + f ( n ) a Ln  1 ( ) nq q q q  (n)    (n)  f (b ) − f ( a )  𝑏𝑏 𝑛𝑛𝑛𝑛 𝑠𝑠  (b − a )  Where 𝑆𝑆(𝑛𝑛, 𝑠𝑠, 𝑞𝑞, 𝑥𝑥) = ∫𝑎𝑎 𝑥𝑥 (𝑥𝑥 − 𝑎𝑎) 𝑑𝑑𝑑𝑑 . ≤ b−a   . (𝑛𝑛) 𝑞𝑞 n!  Proof: If �𝑓𝑓 � for 𝑞𝑞 > 1 is 𝑠𝑠-convex function the first  q n +1 ×  Lnq  (a, b) − aLnq (a, b)  + f ( ) ( a ) Lnq sense on [𝑎𝑎, 𝑏𝑏], using Lemma1.1 and the Hölder integral nq nq 1 + nq     inequality, we have the following inequality: (𝑛𝑛) 𝑞𝑞

(𝑛𝑛) 𝑞𝑞

Turkish Journal of Analysis and Number Theory

Proposition 2.5. Under the conditions of Corollary 2.3 for 𝑛𝑛 = 1 we obtain the inequality

f ( b ) b − f (a)a b−a

1 q

1 f ( x)dx b−a ∫

 f' b q− f' a q ( ) ( ) ≤  b−a 

1

q  Lq +1 (a, b)   q +1  + f ' ( a ) q Lq  . q  −aLq (a, b)  q   

Proposition 2.6. For 𝑞𝑞 = 1 , we obtain the following inequality

f ( b ) b − f (a)a b−a

a+b 1 ( b − a ) Lnnp ( a, b ) f ( n )  . n!  2 

=

a

Corollary 2.4. Under the conditions of Theorem 2.4 for 𝑛𝑛 = 1, we obtain the inequality

f ( b ) b − f (a)a b−a

1 f ( x)dx b−a ∫

Theorem 2.4. For ∀𝑛𝑛 ∈ ℕ ; let f : I ⊂ [ 0, ∞ ) →  be

[2]

n -times differentiable function on I  and a, b ∈ I  with 𝑞𝑞

𝑞𝑞

a < b . If �𝑓𝑓 (𝑛𝑛) � ∈ 𝐿𝐿[𝑎𝑎, 𝑏𝑏] and �𝑓𝑓 (𝑛𝑛) � for 𝑞𝑞 > 1 is 𝑠𝑠-concave function in the first sense on [𝑎𝑎, 𝑏𝑏], then the following inequality holds:

 f ( k ) ( b ) b k +1 − f ( k ) ( a ) a k +1  b  − f ( x)dx 1 − ∑ ( )   ∫ k + 1) ! ( k =0   a

[3] [4]

k

1 n  a+b ≤ ( b − a ) Lnnp ( a, b ) f ( )  , n!  2  where 1/ p + 1/ q = 1.

[5] [6]

𝑞𝑞

Proof: If �𝑓𝑓 (𝑛𝑛) � for 𝑞𝑞 > 1 is 𝑠𝑠-concave function the first sense on [𝑎𝑎, 𝑏𝑏], using Lemma1.1, the Hermite-Hadamard inequality and the Hölder integral inequality, we have the following inequality:

 ( k ) ( b ) b k +1 − f ( k ) ( a ) a k +1  b k f  − f ( x)dx ∑ ( −1)   ∫ + k 1 ! ( ) k =0   a

[7] [8] [9]

n −1

[10]

b

≤

1 n (n) x f ( x ) dx n! ∫

[11]

a

1

b p 1 ≤  ∫x np dx   n!  a  1 p

1  np  x dx   n!  ∫ a  b

≤

1

q b n q  ∫ f ( ) ( x ) dx    a 

[12] [13] 1 q q

 a+b  (b − a ) f ( n )     2      1

 np +1 b  p x 1  f (n)  a + b  = (b − a )     n!  2   np + 1 a    1 q

1

1 = (b − a ) q n!

1  a+b f ( x)dx ≤ L p ( a, b ) f '  . b−a ∫ 2   a

References

a

[1]

n −1

b

−

b

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1 (2b + a ) f ' ( b ) + (2a + b) f ' ( a )  .  6 

≤

1

1 p

 b np +1 − a np +1  p ( n )  a + b  1 = (b − a ) (b − a )   f   n!  2   (np + 1)(b − a ) 

b

−

67

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 b np +1 − a np +1  p ( n )  a + b    f     np + 1  2   

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