GENERALIZATION OF CEBYSEV TYPE INEQUALITIES FOR FIRST DIFFERENTIABLE MAPPINGS

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Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 12 (2011), No. 2, pp. 245–253

A GENERALIZATION OF ˇCEBY ˇSEV TYPE INEQUALITIES FOR

FIRST DIFFERENTIABLE MAPPINGS

ERHAN SET, MEHMET ZEKI SARIKAYA, AND FAROOQ AHMAD

Received February 16, 2011

Abstract. In this paper, we improve and further generalize some ˇCebyˇsev type inequalities in-volving functions whose derivatives belong to Lpspaces via certain integral identities.

2000 Mathematics Subject Classification: 26D07, 26D15, 26D20 Keywords: ˇCebyˇsev type inequalities, Lpspaces

1. INTRODUCTION

In 1882, P. L. ˇCebyˇsev proved the following important integral inequality [13],[9, p.207]: T .f; g/ 1 12.b a/ 2 f0 1 g0 1; (1.1)

where f; g_{W Œa; b ! R are absolutely continuous functions whose derivatives f}0; g0₂ L₁Œa; b and T .f; g/_D 1 b a Z b a f .x/g.x/dx 1 b a Z b a f .x/dx ! 1 b a Z b a g.x/dx ! ; (1.2) which is called the ˇCebyˇsev functional, provided the integrals in (1.2) exist.

During the past few years many researchers have given considerable attention to the inequality of (1.1). Various generalizations, extensions and variants of this in-equality have appeared in the literature, see ([1–13]) and the references cited therein. Recently, Pachpatte [8] established a new ˇCebyˇsev type inequality involving func-tions whose derivatives belong to Lp spaces:

Theorem 1. Letf; gW Œa; b ! R are absolutely continuous functions whose de-rivativesf0; g0_{2 L}pŒa; b, p > 1: Then, we have inequalities

T .f; g/ 1 .b a/3 f0 _p g0 _p Z b a .B.x//2q_dx; c

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and T .f; g/ 1 2 .b a/2 Z b a h g.x/ f0 _pC f .x/ g0 _p i .B.x//q1_dx; _(1.3) where B.x/D.x a/ qC1_{C .b x/}qC1 q_{C 1} ;

forx_{2 Œa; b and} _p1_C1_q_{D 1:}

Motivated by the results of Pachpatte, in the present paper we establish some new ˇ

Cebyˇsev type inequalities for_p1C1q D 1. The analysis used in the proofs is

element-ary and our results provide new estimates for these types of inequalities. 2. MAIN RESULTS

Let Œa; b R, a < bI and as usual for any function h 2 LpŒa; b ; p > 1, we define

khkp D Rb a jh.t/j p dt 1 p

: We use the following notation to simplify the details of presentation. For suitable functions f; gW Œa; b ! R and 2 Œ0; 1 ; we set

T.f; g/ D 1 b a 1 2 Z b a f .x/g.x/dx 4.b a/2 ( Z b a Œ.x b/ f .b/ .x a/ f .a/ g.x/dx C Z b a Œ.x b/ g.b/ .x a/ g.a/ f .x/dx ) 1 b a Z b a f .x/dx ! 1 b a Z b a g.x/dx ! : Now, let us state our main results.

Theorem 2. Letf; g_{W Œa; b ! R be absolutely continuous functions whose} deriv-ativesf0; g0_{2 L}pŒa; b, p > 1: Then, the following inequality holds:

jT.f; g/j 1 2 .b a/2 Z b a h g.x/ f0 _pC f .x/ g0 _p i .Z.x//q1_dx; _(2.1) where Z.x/D .x a/ qC1_{C .b x/}qC1 q_{C 1} 2 qC1 C 1 2 qC1! ; forx_{2 Œa; b ;} _p1_C_q1_{D 1 and 2 Œ0; 1 :}

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Proof. From the hypotheses, we can write the following identities, 1 b a Z b a p.x; t /f0.t /dt D 1 2 f .x/ .x b/ f .b/ .x a/ f .a/ 2.b a/ 1 b a Z b a f .t /dt; (2.2) and 1 b a Z b a p.x; t /g0.t /dt D 1 2 g.x/ .x b/ g.b/ .x a/ g.a/ 2.b a/ 1 b a Z b a g.t /dt; (2.3) for x2 Œa; b ; 2 Œ0; 1 ; where

p.x; t /D 8 ˆ < ˆ : t aC x a 2 ; a t x t b b x 2 ; x < t_b :

A simple proof of equalities of (2.2) and (2.3) can be done by integrating in the left hand side and using the identity of p.x; t /: The details are left for the interested reader.

Multiplying both sides of (2.2) and (2.3) by g.x/ and f .x/ respectively and adding the resulting identities, we get

1 b ag.x/ Z b a p.x; t /f0.t /dt_C 1 b af .x/ Z b a p.x; t /g0.t /dt (2.4) D 2 1 2 f .x/g.x/ .x b/ f .b/ .x a/ f .a/ 2.b a/ g.x/ .x b/ g.b/ .x a/ g.a/ 2.b a/ f .x/ 1 b ag.x/ Z b a f .t /dt 1 b af .x/ Z b a g.t /dt: Multiplying both sides of (2.4) by _{b a}1 ; we get

2 b a 1 2 f .x/g.x/ (2.5) .x b/ f .b/ .x a/ f .a/ 2.b a/2 g.x/ .x b/ g.b/ .x a/ g.a/ 2.b a/2 f .x/ 1 .b a/2g.x/ Z b a f .t /dt 1 .b a/2f .x/ Z b a g.t /dt

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D 1 .b a/2g.x/ Z b a p.x; t /f0.t /dt_C 1 .b a/2f .x/ Z b a p.x; t /g0.t /dt: Integrating both sides of (2.5) with respect to x over Œa; b, we get

2 b a 1 2 Z b a f .x/g.x/dx (2.6) 2.b a/2 ( Z b a Œ.x b/ f .b/ .x a/ f .a/ g.x/dx C Z b a Œ.x b/ g.b/ .x a/ g.a/ f .x/dx ) 2 1 b a Z b a g.x/dx ! 1 b a Z b a f .x/dx ! D 1 .b a/2 Z b a g.x/ Z b a p.x; t /f0.t /dt ! dx C 1 .b a/2 Z b a f .x/ Z b a p.x; t /g0.t /dt ! dx: After the rewriting of the equality of (2.5), we obtain

T.f; g/D 1 2.b a/2 Z b a " g.x/ Z b a p.x; t /f0.t /dt_{C f .x/} Z b a p.x; t /g0.t /dt # dx: (2.7) From (2.7) and using the properties of modulus, H¨older’s integral inequality, we have

jT.f; g/j (2.8) 1 2.b a/2 Z b a " jg.x/j Z b a jp.x; t/j ˇ ˇf0.t /ˇˇdtC jf .x/j Z b a jp.x; t/j ˇ ˇg0.t /ˇˇdt # dx 1 2.b a/2 Z b a 2 4jg.x/j Z b a jp.x; t/j q_dt !1_q Z b a ˇ ˇf0.t /ˇˇ p dt !_p1 C jf .x/j Z b a jp.x; t/j q dt !_q1 Z b a ˇ ˇg0.t / ˇ ˇ p dt !_p13 5dx D 1 2 .b a/2 Z b a h g.x/ f0 _pC f .x/ g0 _p i Z b a jp.x; t/j q dt !_q1 dx:

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A simple calculation shows that Z b a jp.x; t/j q dt (2.9) D Z x a ˇ ˇ ˇt aC x a 2 ˇ ˇ ˇ q dtC Z b x ˇ ˇ ˇ ˇ t b b x 2 ˇ ˇ ˇ ˇ q dt D .x a/q Z x a ˇ ˇ ˇ ˇ t a x a 2 ˇ ˇ ˇ ˇ q dt_{C .b x/}q Z b x ˇ ˇ ˇ ˇ t b b x 2 ˇ ˇ ˇ ˇ q dt D .x a/ qC1_{C .b x/}qC1 qC 1 2 qC1 C 1 2 qC1! D Z.x/:

Using (2.8) and (2.9), we get the required inequality in (2.1). The proof is complete. Remark1. If we choose D 0 in (2.1), then we get (1.3).

Corollary 1. Letf _{W Œa; b ! R be an absolutely continuous function whose} de-rivativef02 LpŒa; b, p > 1: Then, the following inequality holds:

ˇ ˇ ˇ ˇ ˇ 1 2 f .x/ .x b/ f .b/ .x a/ f .a/ 2.b a/ 1 b a Z b a f .t /dt ˇ ˇ ˇ ˇ ˇ 1 b a f0 _p.Z.x// 1 q_: _(2.10)

Proof. From (2.2) using the properties of modulus and applying H¨older’s integral inequalty, we have ˇ ˇ ˇ ˇ ˇ 1 2 f .x/ .x b/ f .b/ .x a/ f .a/ 2.b a/ 1 b a b R a f .t /dt ˇ ˇ ˇ ˇ ˇ 1 b a b R a jf 0_{.t /}_jp dt !_p1 b R ajp.x; t/j q_dt !_q1 D 1 b akf 0_k p.Z.x// 1 q_: Remark2. i) If we choose D 1 in (2.10), then we get

ˇ ˇ ˇ ˇ ˇ 1 2f .x/ .x b/ f .b/ .x a/ f .a/ 2.b a/ 1 b a Z b a f .t /dt ˇ ˇ ˇ ˇ ˇ

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1 2 .b a/ .x a/qC1_{C .b x/}qC1 .q_{C 1/} !1_q f0 _p: ii) For D 0 in (2.10), we have,

ˇ ˇ ˇ ˇ ˇ f .x/ 1 b a Z b a f .t /dt ˇ ˇ ˇ ˇ ˇ _.b1_a/ .x a/ qC1_{C .b x/}qC1 .q_{C 1/} !1_q f0 _p: Theorem 3. Let f; g _{W Œa; b ! R be absolutely continuous functions whose} de-rivativesf0; g0_{2 L Œa; b : If jf}0_jq and_jg0_jq is convex onŒa; b ; p > 1; _p1_C_q1 _{D 1:} Then, the following inequality holds:

jT .f; g/j 1 2 Z b a 8 < : 2 4jf .x/j jg0.a/_jq_{C jg}0.b/_jq 2 1 q C jg.x/j jf 0_.a/_jq C jf0.b/_jq 2 1 q 3 5M 1 p_.x/ 9 = ; dx (2.11) where M.x/_D.b x/ pC1_{C .x a/}pC1 .b a/pC1.p_{C 1/} :

Proof. From the hypotheses, we can write the following identities, f .x/D 1 b a Z b a f .x/dxC .b a/ Z 1 0 P .x; t /f0.t aC .1 t/b/dt; (2.12) and g.x/_D 1 b a Z b a g.x/dx_{C .b a/} Z 1 0 P .x; t /g0.t a_{C .1 t/b/dt;} (2.13) for x_{2 Œa; b ; where}

P .x; t /_D (

t; 0_tb x_{b a} t 1; b x_{b a} < t₁ :

A simple proof of equalities of (2.12) and (2.13) can be obtained by integrating in the right hand side and using the identity of p.x; t /: The details are left for the interested reader.

Multiplying both sides of (2.12) and (2.13) by g.x/ and f .x/ respectively and adding the resulting identities, we get

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D g.x/ b a Z b a f .x/dx_{C .b a/g.x/} Z 1 0 P .x; t /f0.t a_{C .1 t/b/dt} Cf .x/ b a Z b a g.x/dx_{C .b a/f .x/} Z 1 0 P .x; t /g0.t a_{C .1 t/b/dt:} Integrating both sides of (2.14) with respect to x over Œa; b and dividing both sides of the resulting identity .b a/; we get

2 b a Z b a f .x/g.x/dx (2.15) D 2 1 b a Z b a g.x/dx ! 1 b a Z b a f .x/dx ! C Z b a g.x/ Z 1 0 P .x; t /f0.t a_{C .1 t/b/dt} dx C Z b a f .x/ Z 1 0 P .x; t /g0.t aC .1 t/b/dt dx: Rewriting the identity (2.15), we get

T .f; g/ _D 1 2 Z b a g.x/ Z 1 0 P .x; t /f0.t a_{C .1 t/b/dt} dx (2.16) C1 2 Z b a f .x/ Z 1 0 P .x; t /g0.t a_{C .1 t/b/dt} dx: From (2.16), using the properties of modulus and rearranging the terms, we get

jT .f; g/j 1 2 Z b a jf .x/j Z 1 0 jP .x; t/j ˇ ˇg0.t a_{C .1 t/b/}ˇˇdt dx C1 2 Z b a jg.x/j Z 1 0 jP .x; t/j ˇ ˇf0.t a_{C .1 t/b/}ˇ ˇdt dx:

Applying H¨older’s integral inequality and the convexity ofjf0jqandjg0jq; we obtain

jT .f; g/j 1 2 Z b a 2 4jf .x/j Z 1 0 jP .x; t/j p dt 1 p Z 1 0 ˇ ˇg0.t aC .1 t/b/ ˇ ˇ q dt 1 q C jg.x/j Z 1 0 jP .x; t/j p dt 1 pZ 1 0 ˇ ˇf0.t a_{C .1 t/b/}ˇˇ q dt 1 q 3 5dx

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D 1₂ Z b a 8 < : 2 4jf .x/j jg0.a/_jq_{C jg}0.b/_jq 2 1 q (2.17) C jg.x/j jf 0_.a/_jq C jf0.b/jq 2 q1 3 5 Z 1 0 jP .x; t/j p dt 1 p 9 = ; dx:

A simple computation gives Z 1 0 jP .x; t/j p dt _D Z b_b x_a 0 tpdt_C Z 1 b x b a .t 1/pdt (2.18) D .b x/ pC1_{C .x a/}pC1 .b a/pC1.pC 1/ D M.x/:

Using (2.18) in (2.18), we get the required inequality in (2.11). Corollary 2. If in theorem3we require that_jf0.x/_{j K; K > 0 and jg}0.x/_{j N;} N > 0, then the following inequality holds

jT .f; g/j 1 2 Z b a Œjf .x/j N C jg.x/j K Mp1_.x/dx ! ; where _p1_C1_q _{D 1:} REFERENCES

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Authors’ addresses

Erhan Set

Department of Mathematics, Faculty of Science and Arts, D¨uzce University, D¨uzce-TURKEY E-mail address: erhanset@yahoo.com

Mehmet Zeki Sarikaya

Department of Mathematics, Faculty of Science and Arts, D¨uzce University, D¨uzce-TURKEY E-mail address: sarikayamz@gmail.com

Farooq Ahmad

Center for Advanced Studies in Pure and Applied Mathematics, B. Z. U., Multan, Pakistan (Presently at Mathematics Department, Govt. Post Graduate College Bhakkar 30000, Punjab, Pakistan)

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GENERALIZATION OF CEBYSEV TYPE INEQUALITIES FOR FIRST DIFFERENTIABLE MAPPINGS

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