(2) fv*-t)2v(»-«)=(*-a>:+(*-*)2

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1 proceedings of the american mathematical society Volume 23, Number 2, December 995 OSTROWSKI TYPE INEQUALITIES GEORGE A. ANASTASSIOU (Communicated by Andrew M. Bruckner) Abstract. Optimal upper bounds are given to the deviation of a function / 6 CN([a, b]), N 6 N, from its averages. These bounds are of the form A ' H/^'lIco i where A is the smallest universal constant, i.e., the produced inequalities are sharp sometimes are attained. This work has been greatly motivated by the works of Ostrowski (938) Fink (992). 0. Introduction Here we establish optimal upper bounds on the deviation of a function from its averages. These lead to sharp inequalities. Namely, let / Cn+l([a, b]), n Z+, such that ßk)(x) = 0, k = I,..., n, where x is a fixed point in [a, b]. Then we establish that (*) rjj>>*-/w <9n(x)-\\f(n+l)\\ "+)'-oc where <pn(x) is a continuous function that depends only on n, a, b, it has a simple form it is the smallest possible, i.e., (*) is sharp in some cases it is even attained. The special case of x = =^ is encountered.. On Ostrowski's inequality Ostrowski's inequality (see Ostrowski [2]) is as follows: () / f(y)dy-f(x, (*-*?) 4+ ()2 where / e C ([a, b]), x [a, b]. Inequality () is sharp since the function in ( ) cannot be replaced by a smaller one. One can easily notice that (2) fv*-t)2v(»-«)=(*-a>:+(*-*)2 ()2 / v "' 2-() Received by the editors October 3, 993, in revised form, June 7, Mathematics Subject Classification. Primary 26D07, 26D0, 26D5, 4A44; Secondary 26A24, 26D20. Key words phrases. Function average, deviation of a function, best constant, sharp/attained inequality. 995 American Mathematical Society 3775

2 3776 G. A. ANASTASSIOU Next, we give a different proof to () from that of Ostrowski's initial proof of 938 in [2]. Theorem. Let f C ([a, b]), x [a, b]. Then (3) fb f, w ft n ^ f(x-a)2 + (b-x)2\.... I f{y)dy- f(x) < ( 2.{b_a) ) /II Inequality (3) is sharp, namely the optimal function is (4) f*(y):=\y-x\a-(), a >. Proof. Observe that X- [f(y)dy- /(x) = ^ - IjfVöO - f(x))dy So, we have established inequality (3). Note that thus r'(y) = a-\y- *\*hr)'l!^-fw'dy ^w^ym--sy-^dy n/'i ((x-û)2 + (è-x)2). 2 (b - a) x\a~l sign(y -x)-(), \r'(y)\=a.\y-x\a-l-() /*' oc = a. (b - a) (max(è - x, x - a)). Also we notice that /*(x) = 0. Therefore we have for /* that (5) Also, we observe that L.H.S.(3)= / \y-x\a-dy = la lim L.H.S.(3) = a» a+ (x - a)2 + (b - x)2 R.H.S.(3) i.e., proving (3) sharp. (x - a)2 + (b- x)2 ) a (max(z> - x, x - a))a ' lim R.H.S.(3) = a > D (x-a)2 + (6-x)2 lim L.H.S.(3) = lim R.H.S.(3), a» a»

3 OSTROWSKI TYPE INEQUALITIES 3777 Note that when x = a or x = b, inequality (3) can be attained by fa(y) := (y - a)'(b - a), fb(y) := (y - b) (b - a), respectively (then both sides of (3) are equal to (b - a)2/2). 2. More general Ostrowski type inequalities The following material has been greatly motivated by the important work of Fink []. Let / Cn+l([a, b]), n N, x [a, b], be fixed. Then by Taylor's theorem we get (6) f(y) - f(x) = ^ ^ (y -x)k +X (x,y), where (7) %n(x,y) := j\fw(t) - f{n)(x)). iy{~^ dt; here y can be > x or < x. Let y > x ; then i.e., \3ln(x, y)\ < jf \fn)(t) - f-n\x)\ {y~^ <II/^IU7>* -^^ = ll/^iu-^f, dt (8) \^n(x,y)\<^^-(y-x)"+l, y>x. Now let x > v ; then \&n(x, y)\ = \f(f(n)(t) - /«(*)). {y{~^ ~ dt < f\ß"\t)-f("kx)\-ly{-f~y.dt i.e., = M_ü~. (x - V)"+I (n+)! [ y) ' II f(n+l)\\ (9) \&n(x,y)\<u{n + ll~-(x-yy+i, x > y. From (8) (9) we get (0) \Xn{x,y)\<^+^~.\y-x\n+l, for all x, y [a, b].

4 3778 G. A. ANASTASSIOU Next we treat ~f Ay)dy-f(x) " \l y/w(x) ^ k\ k=\ J2^.(y-x)k+^n(x,y) f(f(y)-f(x))-dy dy f (y-x)k-dy+ f Mn(x,y).dy ffm.. \(b- x)*+> _{a_ x)k+i] k=i (k + iy. + f âên(x,y) dy (by (0)) ^b^a k=\ ^ '" \\fin+ + (»+!)! ly ' dy i.e., we have proved that (H) < I ñy)dy-f(x) J2l- ^-\(b-x)k+]-(a-x)k+i> 7c=l (*+l)! + \\f'(n+l) (n + 2)! ((x-a)"+2 + (è-x)n+2) where / e C"+l ([a, b]), n N, x [a, b], is fixed. If we choose x = =^, then (2) bh-samdy-f{ b-x = x-a = a + b 2 ' < /(^(^+) (,-fl)*+i U/t"*')»,,, ( -a)"*2 ^ Ofc + )! 2fc (w + 2)! 2"+!< even<«where /e C"+([a, 6]), " N. The above considerations the established inequalities () (2) lead to the following results.

5 OSTROWSKI TYPE INEQUALITIES 3779 Theorem 2. Let f C"+]([a, b]), n N x [a, b] be fixed, such that fik)(x) = 0, k=l,...,n. Then (3) jf/w-a^iph^-^)- Inequality (3) is sharp. Namely, when n is odd it is attained by f*(y) := (y - x)"+[ '(), while when n is even the optimal function is f(y):=\y-x\"+a-(), «>. Proof. Inequality (3) comes immediately from (). Next we prove the sharpness of inequality (3). When n is odd: Notice that f*(k)(x) = 0, k = 0,,..., n, f*{n+i)(y) = (n + )! -(). Hence Plugging f* into (3) we get ILT("+) oo = («+ l)!-( -a). (4) L.H.S.(3)=( >-^+2 + ^-fl)"+2. K ' K ' n+2 Also, (5, R.H.s.q^'*-'""2-^ -*> ", v ' v ' n+2 From (4) (5), when«is odd, inequality (3) was proved to be sharp, in particular attained by /*. When n is even: Notice that f(k)(x) = 0, k = 0, I,..., n, fin+i\y) = (n + a)(n + a - I) (a + ) a \y - x Q-' sign(y -x) (b - a). Hence \f{n+l](y)\= lfl(n + a-j)y\y-x\a-l-() \\f{n+])\\oo = ( f[(n + a - j) j - (max(è - x, x - a))a~l (b - a). Consequently we have,,., (n;=o(«+ «-7))-(max( -x,x-a))"-' R.H.S.(3) =-^^- ((x - a)n+2 + (b - x)"+2), (6) UmR.H.S.(3)=<^l^±^^ v ' a-\ v ' n + 2 L.H.S.(3)=^-^+Q+'+^-^+Q+. v ; n+a+l a>l.

6 3780 G. A. ANASTASSIOU Therefore (7) liml.h.s.(3) = a» (x - g)"+2 + (b - x)n+2 n + 2 From (6) (7) we get that (3) is sharp also when n is even. D Note that when x = a or x = b n is even, inequality (3) can be attained by fa(y) := (y - a)n+l (b - a), fb(y) := (y-b)n+i-(), respectively (then both sides of (3) are equal to (b - a)n+2/n + 2). When x = (a + b)/2, we have a case of special interest which is described next. Theorem 3. Let f Cn+l([a, b]), nen such that fik>((a + b)/2) = 0,all k even {I,..., n}. Then <\\f{n+l)\\oo ()"+l (n + 2)! ' 2"+' Inequality (8) is sharp. Namely, when n is odd it is attained by f*(y) := (y - =^)"+ '(), while when n is even the optimal function is (8) h-o^-'m f(y) a + b (), a>l. Corollary. Let f C2([a, b]) such that f"((a + b)/2) = 0. Then (9) b i_.rw/(^) < unie ()2 24 which is sharp as in Theorem 3. Proof. Apply Theorem 3 with n =. Proof of Theorem 3. Inequality (8) comes immediately from (2) the assumption f{k)((a + b)/2) = 0, all k even in {,...,«}. Next we prove the sharpness of inequality (8). When n is odd: We notice that r{k) (^y^) =. for A: = 0 all k even e {,. n}, furthermore (20) Also we find (2) /»(i+dq,) = («+ )!. (ft - a), R.H.S.(8) = L.H.S.(8) = (b - a)n+2 («+ 2)-2"+! (b - a)n+2 (n + 2). 2«+'' all y [a, b]. From (20) (2) we get that (8) is attained by /*, therefore (8) has been proved as sharp when n is odd.

7 OSTROWSKI TYPE INEQUALITIES 378 When n is even: We notice that furthermore f(k)((a + b)/2) = 0, for k = 0 all k even in {,...,«}, f {n+l\y) = fl(n +a-j).\y-^\a~l-sign (y-^y () 7=0 ll/("+)lloc = (flin + a - j)) (^f' ' (b- a). {\Tj=o{n + <*-J))'{{b-ä)/2T-i.() * ' A ' (n + 2)! ()n+l HpRT Q>- Hence ()n+2 (22) limr.h.s.(8)= / "\ ^,. v ' Q-i v ' (n + 2)-2"+' Also we find (23) L.H.S.(8)=2'<"'-'"/2r"' v ' n+a+ ()n+2 liml.h.s.(8 =.K", "'... Q-i v ' 2n+i-(n + 2) From (22) (23) we have established that inequality (8) is sharp again when n is even. D References. A. M. Fink, Bounds on the deviation of a function from its averages, Czechoslovak Math. J. 42(7) (992), A. Ostrowski, Über die Absolutabweichung einer differentiebaren Funktion von ihrem Integralmittelwert, Comment. Math. Helv. 0 (938), Department of Mathematical Sciences, The University of Memphis, Memphis, Tennessee 3852