Int. J. Anal. Appl. (2022), 20:16 Weighted Ostrowski’s Type Integral Inequalities for Mapping Whose First Derivative Is Bounded S. Fahad1, M. A. Mustafa2, Z. Ullah3, T. Hussain2, A. Qayyum2,∗ 1Bahauddin Zakriya University Multan-Pakistan 2Institute of Southern Punjab Multan-Pakistan 3Department of Mathematics, Division of Science and Technology, University of Education Lahore-Pakistan ∗Corresponding author: atherqayyum@isp.edu.pk Abstract. The aim of paper is to develop the inequalities for L∞, Lp and L1 norms. Applications for some special weight functions and Perturbed expressions are also determined via Chebychev functional. We recaptured the previous results for different weights. 1. Introduction In 1938, Ostrowski established the interesting integral inequality for differentiable mappings with bounded derivative [10]. Cerone [3] also worked on this inequality. Different authors worked on the generalization of Ostrowski’s type inequalities that is [1]- [2] and [9]. Further work done by Iftikhar et al. [6], Mustafa et al. [7] and Qayyum et al. [12]- [14]. Let the functional S ( f ; $; Ĵ, ǩ ) be defined as: S ( f ; $; Ĵ, ǩ ) = f (z̈) −M̈ ( f ; $; Ĵ, ǩ ) , (1.1) where f (z̈) : [ Ĵ, ǩ ] →R be a continuous mapping, M̈ ( f ; $; Ĵ, ǩ ) is weighted integral mean and is defined as: M̈ ( f ; $; Ĵ, ǩ ) = 1 ǩ − Ĵ ∫ ǩ Ĵ f (ř) $ (ř) dř. (1.2) The functional S ( f ; $; Ĵ, ǩ ) represents the deviation of f (z̈) from its integral mean over [ Ĵ, ǩ ] . Received: Jan. 12, 2022. 2010 Mathematics Subject Classification. 35A23. Key words and phrases. Ostrowski inequality; weight function; numerical integration. https://doi.org/10.28924/2291-8639-20-2022-16 ISSN: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-16 2 Int. J. Anal. Appl. (2022), 20:16 We assume non-negative weight function $ : (Ĵ, ǩ) → [0,∞) is integrable∫ ǩ Ĵ $(ř)dř < ∞. (1.3) We define m, m1 and µ as m ( Ĵ, ǩ ) = ∫ ǩ Ĵ $(ř)dř, m1 ( Ĵ, ǩ ) = ∫ ǩ Ĵ ř$(ř)dř and µ ( Ĵ, ǩ ) = m1 ( Ĵ, ǩ ) m ( Ĵ, ǩ ) . (1.4) 2. Main Result Theorem 2.1. Let f : [ Ĵ, ǩ ] → R be continuous on [ Ĵ, ǩ ] and differentiable mapping on ( Ĵ, ǩ ) , then the following weighted peano kernel, define Ġ (., .) : [ Ĵ, ǩ ] →R as: Ġ (z̈, ř) =   � (�+δ)(z̈−Ĵ) ∫ ř Ĵ $ (u) du, if ř ∈ [ Ĵ, z̈ ] δ (�+δ)(ǩ−z̈) ∫ ř ǩ $ (u) du, if ř ∈ ( z̈, ǩ ] (2.1) ∀ ř ∈ [ Ĵ, ǩ ] , z̈ ∈ [ Ĵ, ǩ ] , $ is weight function as stated in (1.3) and �,δ ∈ R non-negative and both are not zero at a time. Then the following weighted integral identity τ ($; z̈; �,δ) = ∫ ǩ Ĵ Ġ (z̈, ř) f ′ (ř) dř = Bf (z̈) − 1 � + δ [ �M̈ ( f ; $; Ĵ, z̈ ) + δM̈ ( f ; $; z̈, ǩ )] , (2.2) holds, where B = 1 � + δ [ � z̈ − Ĵ m ( Ĵ, z̈ ) + δ ǩ − z̈ m ( z̈, ǩ )] , M̈ ( f ; $; Ĵ, ǩ ) is weighted integral mean as defined in (1.2). Proof. From (2.1), we have∫ ǩ Ĵ Ġ (z̈, ř) f ′ (ř) dř = 1 � + δ { � z̈ − Ĵ ∫ z̈ Ĵ $ (ř) dř + δ ǩ − z̈ ∫ ǩ z̈ $ (ř) dř } f (z̈) − 1 � + δ { � z̈ − Ĵ ∫ z̈ Ĵ f (ř) $ (ř) dř + δ ǩ − z̈ ∫ ǩ z̈ f (ř) $ (ř) dř } , where the integration by parts formula has been utilized on the separate interval [ Ĵ, z̈ ] and ( z̈, ǩ ] . Simplification of the expressions readily produces the identity as stated in (2.2). � Int. J. Anal. Appl. (2022), 20:16 3 Theorem 2.2. Let f : [ Ĵ, ǩ ] →R be continuous on [ Ĵ, ǩ ] and differentiable mapping on ( Ĵ, ǩ ) , whose first derivative f ′ : [ Ĵ, ǩ ] → R is bounded on ( Ĵ, ǩ ) , then following weighted integral inequalities |τ ($; z̈; �,δ)| ≤   ( �m(Ĵ,z̈) z̈−Ĵ { z̈ −µ ( Ĵ, z̈ )} + δm(z̈,ǩ) ǩ−z̈ { z̈ −µ ( z̈, ǩ )}) ∥∥∥f ′∥∥∥ ∞ �+δ for f ′ ∈ L∞ [ Ĵ, ǩ ] ∥∥∥f ′∥∥∥ p $(z̈) (�+δ)(q̈+1) 1 q̈ [ �q̈ ( z̈ − Ĵ ) + δq̈ ( ǩ − z̈ )]1 q̈ for f ′ ∈ Lp [ Ĵ, ǩ ] ϑ �+δ [ 1 + |ρ| ϑ ] ∥∥∥f ′∥∥∥ 1 2 for f ′ ∈ L1 [ Ĵ, ǩ ] (2.3) are hold for all ř ∈ [ Ĵ, ǩ ] , z̈ ∈ [ Ĵ, ǩ ] , $ is weight function as stated in (1.3) and �,δ ∈R non-negative and both are not zero at a time, where ϑ = 1( z̈ − Ĵ )( ǩ − z̈ ) (�m(Ĵ, z̈)(ǩ − z̈) + δm(z̈, ǩ)(z̈ − Ĵ)) and ρ = 1( z̈ − Ĵ )( ǩ − z̈ ) (�m(Ĵ, z̈)(ǩ − z̈)−δm(z̈, ǩ)(z̈ − Ĵ)) . Proof. Taking the modulus of (2.2) and using (1.2) |τ ($; z̈; �,δ)| = ∣∣∣∣∣ ∫ ǩ Ĵ Ġ (z̈, ř) f ′ (ř) dř ∣∣∣∣∣ ≤ ∫ ǩ Ĵ ∣∣Ġ (z̈, ř)∣∣ ∣∣∣f ′ (ř)∣∣∣dř, (2.4) where we use properties of the integral and modulus. Thus for f ′ ∈ L∞ [ Ĵ, ǩ ] from (2.4) |τ ($; z̈; �,δ)| ≤ ∥∥∥f ′∥∥∥ ∞ ∫ ǩ Ĵ ∣∣Ġ (z̈, ř)∣∣dř from which a simple calculation using (2.1), gives∫ ǩ Ĵ Ġ (z̈, ř) dř = 1 � + δ [ � z̈ − Ĵ { z̈m ( Ĵ, z̈ ) −m1 ( Ĵ, z̈ )} + δ ǩ − z̈ { z̈m ( z̈, ǩ ) −m1 ( z̈, ǩ )}] . From above, first inequality given in (2.3) is obtained. 4 Int. J. Anal. Appl. (2022), 20:16 Further, using Hölder’s Inequality, we have for f ′ ∈ Lp [ Ĵ, ǩ ] from (2.4) |τ ($; z̈,�,δ)| ≤ ∥∥∥f ′∥∥∥ p (∫ ǩ Ĵ ∣∣Ġ (z̈, ř)∣∣q̈ dř )1 q̈ , where 1 p + 1 q̈ = 1, p > 1. With the help of mean value theorem and by using the technique Qayyum et al. [11], we get (∫ ǩ Ĵ Ġ |(z̈, ř)|q̈ dř )1 q̈ = $ (z̈) (� + δ) (q̈ + 1) 1 q̈ [ �q̈ ( z̈ − Ĵ ) − (−1)q̈+1 δq̈ ( ǩ − z̈ )]1q̈ . So the second inequality given in (2.3) is obtained. Finally, for f ′ ∈ L1 [ Ĵ, ǩ ] we have from (2.4) and using (2.1) |τ ($; z̈; �,δ)| ≤ sup ř∈[Ĵ,ǩ] ∣∣Ġ (z̈, ř)∣∣∥∥∥f ′∥∥∥ 1 , where sup ř∈[Ĵ,ǩ] ∣∣Ġ (z̈, ř)∣∣ = 1 � + δ max ( � z̈ − Ĵ m ( Ĵ, z̈ ) , δ ǩ − z̈ m ( z̈, ǩ )) = 1 2 (� + δ) ( z̈ − Ĵ )( ǩ − z̈ ) [�m(Ĵ, z̈)(ǩ − z̈) +δm ( z̈, ǩ )( z̈ − Ĵ )] ×  1 + ∣∣∣∣ 1(z̈−Ĵ)(ǩ−z̈) [�m(Ĵ, z̈)(ǩ − z̈)−δm(z̈, ǩ)(z̈ − Ĵ)] ∣∣∣∣ 1 (z̈−Ĵ)(ǩ−z̈) [ �m ( Ĵ, z̈ )( ǩ − z̈ ) + δm ( z̈, ǩ )( z̈ − Ĵ )]   . Hence proved. � Remark 2.1. Triangular Inequality from (2.2) and (1.1) is (� + δ) τ ($; z̈; �,δ) = �S ( f ; $; Ĵ, z̈ ) + δS ( f ; $; z̈, ǩ ) Int. J. Anal. Appl. (2022), 20:16 5 then using triangular inequality in (2.3), we get |(� + δ) τ ($; z̈; �,δ)| ≤   � 2 ( m(Ĵ,z̈) z̈−Ĵ { z̈ −µ ( Ĵ, z̈ )})∥∥f ′∥∥∞,[Ĵ,z̈] + δ 2 ( m(z̈,ǩ) ǩ−z̈ { z̈ −µ ( z̈, ǩ )})∥∥f ′∥∥∞,[z̈,ǩ] for f ′ ∈ L∞ [ Ĵ, ǩ ] �$(z̈) ( z̈−Ĵ q̈+1 )1 q̈ ∥∥f ′∥∥ p,[Ĵ,z̈] +δ$(z̈) ( ǩ−z̈ q̈+1 )1 q̈ ∥∥f ′∥∥ p,[z̈,ǩ] for f ′ ∈ Lp [ Ĵ, ǩ ] ϑ 2 ∥∥f ′∥∥ 1,[Ĵ,z̈] + |ρ| 2 ∥∥f ′∥∥ 1,[z̈,ǩ] for f ′ ∈ L1 [ Ĵ, ǩ ] . (2.5) Remark 2.2. Since we may write (2.2) as �M̈ ( f ; $; Ĵ, z̈ ) + δM̈ ( f ; $; z̈, ǩ ) = �M̈ ( f ; $; Ĵ, z̈ ) + δ ǩ − z̈ (∫ ǩ Ĵ $ (ř) f (ř) dř − ∫ z̈ Ĵ $ (ř) f (ř) dř ) = [ �−δ ( z̈ − Ĵ ǩ − z̈ )] M̈ ( f ; $; Ĵ, z̈ ) + δ ( ǩ − Ĵ ǩ − z̈ ) M̈ ( f ; $; z̈, ǩ ) , thus τ ($; z̈; �,δ) is 1 B τ ($; z̈; �,δ) = f (z̈) − 1 B [( 1 − δ � + δ λ ) M̈ ( f ; $; Ĵ, z̈ ) + δ � + δ λM̈ ( f ; $; z̈, ǩ )] , where λ = ǩ − Ĵ ǩ − z̈ , same as [ Ĵ, ǩ ] , M̈ ( f ; $; Ĵ, ǩ ) is also fixed. 6 Int. J. Anal. Appl. (2022), 20:16 Corollary 2.1. Let the conditions of Theorem 2.2 holds. Then the results for δ = � |τ ($; z̈; �,�)| ≤   ( m(Ĵ,z̈) z̈−Ĵ { z̈ −µ ( Ĵ, z̈ )} + m(z̈,ǩ) ǩ−z̈ { z̈ −µ ( z̈, ǩ )}) ∥∥∥f ′∥∥∥ ∞ 2 for f ′ ∈ L∞ [ Ĵ, ǩ ] ( ǩ−Ĵ q̈+1 )1q̈ ∥∥∥f ′∥∥∥ p $(z̈) 2 for f ′ ∈ Lp [ Ĵ, ǩ ] ζ [ 1 + |η| ζ ] ∥∥∥f ′∥∥∥ 1 4 for f ′ ∈ L1 [ Ĵ, ǩ ] (2.6) where τ ($; z̈; �,�) := 1 2 [( 1 z̈ − Ĵ m ( Ĵ, z̈ ) + 1 ǩ − z̈ m ( z̈, ǩ )) f (z̈) − { M̈ ( f ; $; Ĵ, z̈ ) + M̈ ( f ; $; z̈, ǩ )}] , ζ = 1( z̈ − Ĵ )( ǩ − z̈ ) [m(Ĵ, z̈)(ǩ − z̈) + m(z̈, ǩ)(z̈ − Ĵ)] and η = 1( z̈ − Ĵ )( ǩ − z̈ ) [m(Ĵ, z̈)(ǩ − z̈)−m(z̈, ǩ)(z̈ − Ĵ)] . Proof. The result is readily obtained on allowing � = δ in (2.3) so that the left hand side is τ ($; z̈; �,�) from (2.4). � Corollary 2.2. According to Theorem 2.2, then mid point ( z̈ = Ď ⇒ Ĵ+ǩ 2 ) inequality from (2.2)∣∣τ ($; Ď,�,δ)∣∣ ≤   2 ǩ−Ĵ [ �m ( Ĵ, Ď ){ Ď−µ ( Ĵ, Ď )} +δm ( Ď, ǩ ){ Ď−µ ( Ď, ǩ )}] ∥∥∥f ′∥∥∥ ∞ �+δ for f ′ ∈ L∞ [ Ĵ, ǩ ] [ �q̈ + δq̈ ]1 q̈ ( ǩ−Ĵ 2(q̈+1) )1 q̈ ∥∥∥f ′∥∥∥ p $(Ď) (�+δ) for f ′ ∈ Lp [ Ĵ, ǩ ] % (�+δ) [ 1 + |Ψ| % ] ∥∥∥f ′∥∥∥ 1 2 for f ′ ∈ L1 [ Ĵ, ǩ ] (2.7) where % = 1 ǩ − Ĵ [ �m ( Ĵ, Ď ) + δm ( Ď, ǩ )] Int. J. Anal. Appl. (2022), 20:16 7 and Ψ = 1 ǩ − Ĵ [ �m ( Ĵ, Ď ) −δm ( Ď, ǩ )] . Proof. Placing z̈ = Ď ⇒ Ĵ+ǩ 2 in (2.4) and (2.3) produces the results as stated in (2.7). � Corollary 2.3. When the conditions of Theorem 2.2 hold and � = δ using in (2.7) is evaluated at mid point ( z̈ = Ď ⇒ Ĵ+ǩ 2 ) ∣∣τ ($; Ď,�,�)∣∣ ≤   [ m ( Ĵ, Ď ){ Ď−µ ( Ĵ, Ď )} +m ( Ď, ǩ ){ Ď−µ ( Ď, ǩ )}] ∥∥∥f ′∥∥∥ ∞ ǩ−Ĵ for f ′ ∈ L∞ [ Ĵ, ǩ ] ( ǩ−Ĵ q̈+1 )1q̈ ∥∥∥f ′∥∥∥ p $(Ď) 2 for f ′ ∈ Lp [ Ĵ, ǩ ] ζ 2 [ 1 + |η| ζ ] ∥∥∥f ′∥∥∥ 1 2 for f ′ ∈ L1 [ Ĵ, ǩ ] (2.8) where ζ = 2( ǩ − Ĵ ) [m(Ĵ, Ď) + m(Ď, ǩ)] and η = 2( ǩ − Ĵ ) [m(Ĵ, Ď)−m(Ď, ǩ)] . Proof. Putting � = δ in (2.7), we get (2.8). � Remark 2.3. For $ (z̈) = 1 in (2.3) and (2.5) − (2.8), we get Cerone’s results [3]. 3. Application for Some Special Means Now we discuss application for some special means by taking different weights. Remark 3.1. For Uniform (Legendre) mean: Let $ (ř) = 1 put in (2.3) and in (2.4), we get Cerone’s results [3]. Remark 3.2. For Logarithm mean: Let $ (ř) = ln (1/ř) , Ĵ = 0, ǩ = 1, then µ ( Ĵ, ǩ ) is µ (0, 1) = 1 4 , 8 Int. J. Anal. Appl. (2022), 20:16 then ∣∣∣∣ [ 1 � + δ { � z̈ − Ĵ + δ ǩ − z̈ }]( f (z̈) − ∫ 1 0 ln (1/ř) f (ř) dř )∣∣∣∣ ≤   [ 1 �+δ { � z̈−Ĵ + δ ǩ−z̈ }]( z̈ − 1 4 )∥∥f ′∥∥∞ for f ′ ∈ L∞ [Ĵ, ǩ] ∥∥∥f ′∥∥∥ p ln(1/ř) (�+δ)(q̈+1) 1 q̈ [ �q̈ ( z̈ − Ĵ ) + δq̈ ( ǩ − z̈ )]1 q̈ for f ′ ∈ Lp [ Ĵ, ǩ ] || f ′′||1 2(�+δ) ( � z̈−Ĵ + δ ǩ−z̈ )∣∣∣ � z̈−Ĵ − δ ǩ−z̈ ∣∣∣ for f ′ ∈ L1 [Ĵ, ǩ] . holds. Remark 3.3. For Jacobi mean: Let $ (ř) = 1/ √ ř, Ĵ = 0, ǩ = 1, in (1.4), we get µ (0, 1) = 1 3 , then ∣∣∣∣ [ 1 � + δ { � z̈ − Ĵ + δ ǩ − z̈ }]( 2f (z̈) − ∫ 1 0 f (ř) 1/ √ řdř )∣∣∣∣ ≤   [ 2 �+δ { � z̈−Ĵ + δ ǩ−z̈ }]( z̈ − 1 3 )∥∥f ′∥∥∞ for f ′ ∈ L∞ [Ĵ, ǩ] ∥∥∥f ′∥∥∥ p 1/ √ ř (�+δ)(q̈+1) 1 q̈ [ �q̈ ( z̈ − Ĵ ) + δq̈ ( ǩ − z̈ )]1 q̈ for f ′ ∈ Lp [ Ĵ, ǩ ] || f ′′||1 (�+δ) ( � z̈−Ĵ + δ ǩ−z̈ )∣∣∣ � z̈−Ĵ − δ ǩ−z̈ ∣∣∣ for f ′ ∈ L1 [Ĵ, ǩ] . Remark 3.4. For Chebyshev mean: Let $ (ř) = 1/ √ 1 − ř2, Ĵ = −1, ǩ = 1, then µ (−1, 1) = 0, Int. J. Anal. Appl. (2022), 20:16 9 thus ∣∣∣∣ 1� + δ { � z̈ − Ĵ + δ ǩ − z̈ }( πf (z̈) − ∫ 1 −1 1/ √ 1 − ř2f (ř) dř )∣∣∣∣ ≤   1 �+δ { � z̈−Ĵ + δ ǩ−z̈ } (πz̈) ∥∥f ′∥∥∞ for f ′ ∈ L∞ [Ĵ, ǩ] ∥∥∥f ′∥∥∥ p 1/ √ 1−ř2 (�+δ)(q̈+1) 1 q̈ [ �q̈ ( z̈ − Ĵ ) + δq̈ ( ǩ − z̈ )]1 q̈ for f ′ ∈ Lp [ Ĵ, ǩ ] π|| f ′′||1 2(�+δ) ( � z̈−Ĵ + δ ǩ−z̈ )∣∣∣ � z̈−Ĵ − δ ǩ−z̈ ∣∣∣ for f ′ ∈ L1 [Ĵ, ǩ] . Remark 3.5. For Laguerre mean: Let $ (ř) = e−ř Ĵ = 0, ǩ = ∞, then µ (0,∞) = 1, and ∣∣∣∣ 1� + δ { � z̈ − Ĵ + δ ǩ − z̈ }( f (z̈) − ∫ ∞ 0 e−řf (ř) dř )∣∣∣∣ ≤   1 �+δ { � z̈−Ĵ + δ ǩ−z̈ } {z̈ − 1} ∥∥f ′∥∥∞ for f ′ ∈ L∞ [Ĵ, ǩ] ∥∥∥f ′∥∥∥ p e−ř (�+δ)(q̈+1) 1 q̈ [ �q̈ ( z̈ − Ĵ ) + δq̈ ( ǩ − z̈ )]1 q̈ for f ′ ∈ Lp [ Ĵ, ǩ ] || f ′′||1 2(�+δ) ( � z̈−Ĵ + δ ǩ−z̈ )∣∣∣ � z̈−Ĵ − δ ǩ−z̈ ∣∣∣ for f ′ ∈ L1 [Ĵ, ǩ] . holds. Remark 3.6. For Hermite mean: Let $ (ř) = e−ř 2 Ĵ = −∞, ǩ = ∞, then µ (−∞,∞) = 0, 10 Int. J. Anal. Appl. (2022), 20:16 and ∣∣∣∣ 1� + δ { � z̈ − Ĵ + δ ǩ − z̈ }( √ πf (z̈) − ∫ ∞ −∞ e−ř 2 f (ř) dř )∣∣∣∣ ≤   1 �+δ { � z̈−Ĵ + δ ǩ−z̈ }(√ πz̈ )∥∥f ′∥∥∞ for f ′ ∈ L∞ [Ĵ, ǩ] ∥∥∥f ′∥∥∥ p e−ř 2 (�+δ)(q̈+1) 1 q̈ [ �q̈ ( z̈ − Ĵ ) + δq̈ ( ǩ − z̈ )]1 q̈ for f ′ ∈ Lp [ Ĵ, ǩ ] √ π|| f ′′||1 2(�+δ) ( � z̈−Ĵ + δ ǩ−z̈ )∣∣∣ � z̈−Ĵ − δ ǩ−z̈ ∣∣∣ for f ′ ∈ L1 [Ĵ, ǩ] . 4. Perturbed Results For Weighted Ostrowski Type Inequalities Perturbed versions of the results of the previous section may be obtained by using Grüss type results involving Chebychev functional Ť (f ,g; $) = M̈ (f g; $) −M̈ (f ; $) M̈ (g; $) , (4.1) where M̈ (f ; $) is the weighted integral mean as defined in (1.2). For f ,g : [ Ĵ, ǩ ] −→R and integrable on [ Ĵ, ǩ ] , as is their product, then ∣∣Ť (f ,g)∣∣ ≤ Ť 1 2 (f , f ) Ť 1 2 (g,g) , Dragomir [4] for f , g ∈ L2 [ Ĵ, ǩ ] ≤ Γ −γ 2 ř 1 2 (f , f ) , Matic et al. [8] for γ ≤ g (ř) ≤ Γ, ř ∈ [ Ĵ, ǩ ] ≤ (Γ −γ) (Φ −φ) 4 , Grüss [5] for φ ≤ g (ř) ≤ Ψ, ř ∈ [ Ĵ, ǩ ] . (4.2) We obtain following theorem: Theorem 4.1. Let f : [ Ĵ, ǩ ] −→R be an absolutely continuous mapping and � ≥ 0, δ ≥ 0, � + δ 6= 0, then ∣∣∣∣τ ($; z̈; �,δ) − (z̈ −γ)2 S ∣∣∣∣ ≤ ( ǩ − Ĵ ) κ (z̈) [ 1 ǩ − Ĵ ∥∥∥f ′∥∥∥2 2 −S2 ]1 2 , f ′ ∈ L2 [ Ĵ, ǩ ] ≤ ( ǩ − Ĵ ) κ (z̈) Γ −γ 2 , γ ≤ f ′ (ř) ≤ Γ, ř ∈ [ Ĵ, ǩ ] ≤ ( ǩ − Ĵ ) Γ −γ 4 . (4.3) Int. J. Anal. Appl. (2022), 20:16 11 The constant 1 4 is the best possible, where τ ($; z̈; �,δ) is as given in (2.2), γ = �Ĵ + δǩ � + δ , S = f ( ǩ ) − f ( Ĵ ) ǩ − Ĵ , (4.4) κ2 = $ (z̈) 2 [ 1 3 (( � � + δ )2 ( z̈ − Ĵ ) + ( δ � + δ )2 ( ǩ − z̈ )) − ( (z̈ −γ) 2 ( ǩ − Ĵ ) )2 . (4.5) Proof. Associating f (ř) with Ġ (z̈, ř) and g (ř) with f ′ (ř), then from (2.1) and (4.1) , we get Ť ( Ġ (z̈, .) , f ′ (.) ) = M̈ ( Ġ (z̈, .) , f ′ (.) ) −M̈ ( Ġ (z̈, .) ) M̈ ( , f ′ (.) ) . By using (2.1) ( ǩ − Ĵ ) Ť ( Ġ (z̈, .) , f ′ (.) ) (4.6) = τ ($; z̈; �,δ) − ( ǩ − Ĵ ) M̈ ( Ġ (z̈, .) ) S. Now from (2.1) ( ǩ − Ĵ ) M̈ ( Ġ (z̈, .) ) = ∫ ǩ Ĵ Ġ (z̈, ř) dř = $ (z̈) � + δ [ � z̈ − Ĵ ( z̈ − Ĵ )2 2 − δ ǩ − z̈ ( ǩ − z̈ )2 2 ] = $ (z̈) 2 (z̈ −γ) (4.7) (4.7) and (4.5) gives the left hand side of (4.3). Now, for the bounds on (4.6) from (4.2), we have to find Ť 1 2 ( Ġ (z̈, .) , Ġ (z̈, .) ) and φ ≤ Ġ (z̈, .) ≤ Φ. Firstly, we note however that 0 ≤ Ť 1 2 ( f ′ (.) , f ′ (.) ) = [ M̈ ( f ′ (.) )2 −M̈2 ( f ′ (.) )]12 =   1 ǩ − Ĵ ∫ ǩ Ĵ [ f ′ (ř) ]2 dř − ( 1 ǩ − Ĵ ∫ ǩ Ĵ f ′ (ř) dř )2 1 2 = [ 1 ǩ − Ĵ ∥∥∥f ′ (ř)∥∥∥2 2 −S2 ]1 2 ≤ Γ −γ 2 , where γ ≤ f ′ (ř) ≤ Γ, ř ∈ [ Ĵ, ǩ ] . (4.8) 12 Int. J. Anal. Appl. (2022), 20:16 Now from (2.1), the definition of Ġ (z̈, ř), we have κ (z̈) 2 = Ť ( Ġ (z̈, .) , Ġ (z̈, .) ) = M̈ ( Ġ2 (z̈, .) ) −M̈2 ( Ġ (z̈, .) ) , (4.8-1) from (4.7) M̈ ( Ġ (z̈, .) ) = $ (z̈) (z̈ −γ) 2 ( ǩ − Ĵ ) and M̈ ( Ġ2 (z̈, .) ) = ( � (� + δ) ( z̈ − Ĵ ) )2 ∫ z̈ Ĵ (∫ ř Ĵ $ (u) du )2 dř + ( δ (� + δ) ( ǩ − z̈ ))2 ∫ ǩ z̈ (∫ ř ǩ $ (u) du )2 dř = $ (z̈) 2 3 [( � � + δ )2 ( z̈ − Ĵ ) + ( δ � + δ )2 ( ǩ − z̈ )] . By substituting the derived results into (4.8-1), gives 0 ≤ κ (z̈) = Ť 1 2 ( Ġ (z̈, .) , Ġ (z̈, .) ) , (4.9) which is given explicitly by (4.5). We observe from (2.1), that for �,δ ≥ 0 and both are not zero at a time give Φ = sup ř∈[Ĵ,ǩ] Ġ (z̈, ř) and φ = inf ř∈[Ĵ,ǩ] Ġ (z̈, ř) , giving Φ = � �+δ and φ = δ �+δ . Hence, proved the result. � Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication of this paper. References [1] N. S. Burnett, P. Cerone, S. S. Dragomir, J. Roumeliotis, A. Sofo, a Survey on Ostrowski Type Inequalities for Twice Differentiable Mappings and Applications, Inequal. Theory Appl. 1 (2001), 24-30. [2] H. Budak, M. Z. Sarikaya, A. Qayyum, New Refinements and Applications of Ostrowski Type Inequalities for Mappings Whose nth Derivatives Are of Bounded Variation, TWMS J. Appl. Eng. Math. 11 (2021), 424-435. [3] P. Cerone, A New Ostrowski Type Inequality Involving Integral Means Over End Intervals, Tamkang J. Math. 33 (2002), 109–118. https://doi.org/10.5556/j.tkjm.33.2002.290. [4] S. S. Dragomir, Better Bounds in Some Ostrowski-Gruss Type Inequalities, Tamkang J. Math. 32 (2001), 211–216. https://doi.org/10.5556/j.tkjm.32.2001.376. [5] D. S. Mitrinović, J. E. Pečarić, A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993. https://doi.org/10.5556/j.tkjm.33.2002.290 https://doi.org/10.5556/j.tkjm.32.2001.376 Int. J. Anal. Appl. (2022), 20:16 13 [6] M. Iftikhar, A. Qayyum, S. Fahad, M. Arslan, A New Version of Ostrowski Type Integral Inequalities for Different Differentiable Mapping, Open J. Math. Sci. 5 (2021), 353–359. https://doi.org/10.30538/oms2021.0170. [7] M. A. Mustafa, A. Qayyum, T. Hussain, M. Saleem, Some Integral Inequalities for the Quadratic Functions of Bounded Variations and Application, Turk. J. Anal. Numb. Theory 10 (2022), 1–3. https://doi.org/10.12691/ tjant-10-1-1. [8] M. Matic, J. E. Pečarić, and N. Ujević, On New Estimation of the Remainder in Generalized Taylor’s Formula, Math. Inequal. Appl. 2 (1999), 343-361. [9] J. Nasir, S. Qaisar, S. I. Butt, A. Qayyum, Some Ostrowski Type Inequalities for Mappings Whose Second Derivatives Are Preinvex Function via Fractional Integral Operator, AIMS Math. 7 (2022), 3303–3320. https://doi.org/10. 3934/math.2022184. [10] A. Ostrowski, Uber die Absolutabweichung einer differentienbaren Funktionen von ihren Integralmittelwert, Com- ment. Math. Helv. 10 (1938), 226–227. [11] A. Qayyum, A Weighted Ostrowski Gruss Type Inequality for Twice Differentiable Mappings and Applications, Int. J. Math. Comput. 1 (2008), 63-71. [12] A. Qayyum, M. Shoaib, M. Amer latif, A Generalized Inequality of Ostrowski Type for Twice Differentiable Bounded Mappings and Applications, Appl. Math. Sci. 8 (2014), 1889-1901. [13] A. Qayyum, I. Faye, M. Shoaib, M. A. Latif, A Generalization of Ostrowski Type Inequality for Mappings Whose Second Derivatives Belong to L 1 (Ĵ, ǩ) and Applications, Int. J. Pure Appl. Math. Sci. 98 (2015), 169-180. [14] A. Qayyum, A. R. Kashif, M. Shoaib, I. Faye, Derivation and Applications of Inequalities of Ostrowski Type for n-Times Differentiable Mappings for Cumulative Distribution Function and Some Quadrature Rules, J. Nonlinear Sci. Appl. 9 (2016), 1844-1857. https://doi.org/10.30538/oms2021.0170 https://doi.org/10.12691/tjant-10-1-1 https://doi.org/10.12691/tjant-10-1-1 https://doi.org/10.3934/math.2022184 https://doi.org/10.3934/math.2022184 1. Introduction 2. Main Result 3. Application for Some Special Means 4. Perturbed Results For Weighted Ostrowski Type Inequalities References