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1 Available olie at J. Math. Comput. Sci. 2 (202, No. 3, ISSN: ON PARAMETER DEPENDENT REFINEMENT OF DISCRETE JENSEN S INEQUALITY FOR OPERATOR CONVEX FUNCTIONS L. HORVÁTH, KHURAM ALI KHAN 2,3,, AND J. PEČARIĆ2,4 Departmet of Mathematics, Uiversity of Paoia, Egyetem u.0, 8200 Veszprém, Hugary 2 Abdus Salam School of Mathematical Scieces, GC Uiversity, 68-B, New Muslim Tow, Lahore 54600, Pakista 3 Departmet of mathematics, Uiversity of Sargodha, Sargodha 4000, Pakista 4 Faculty of Textile Techology, Uiversity of Zagreb, 0000 Zagreb, Croatia Abstract. I this paper, we cosider the class of self-adjoit operators defied o a Hilbert space, whose spectra are cotaied i a iterval. We give parameter depedet refiemet of the well kow discrete Jese s iequality i this class. The parameter depedet mixed symmetric meas are defied for a subclass of positive self-adjoit operators which isure the refiemets of iequality betwee power meas of strictly positive operators. Keywords: self-adjoit operators, operator covex fuctios, operator meas, symbolic calculus. 200 AMS Subject Classificatio: 26D5; 47A63; 47A64; 47B5. Correspodig author addresses: lhorvath@almos.ui-pao.hu (L. Horváth, khuramsms@gmail.com (K. A. Kha, pecaric@mahazu.hazu.hr (J. Pečarić This research was partially fuded by Higher Educatio Commissio, Pakista. The research, of st author was supported by Hugaria Natioal Foudatios for Scietific Research Grat No. K027, ad of 3rd author was supported by the Croatia Miistry of Sciece, Educatio ad Sports uder the Research Grat Received February 9,

2 ON PARAMETER DEPENDENT REFINEMENT OF DISCRETE JENSEN S INEQUALITY Itroductio ad Prelimiary Results Iitially a complex Hilbert space H is give. The Baach algebra of all bouded liear operators o H is deoted by B(H. Sp(A meas the spectrum of the operator A B(H. Let S(I be the class of all self-adjoit bouded operators o H whose spectra are cotaied i a iterval I R. A fuctio f : D f ( R R is operator mootoe o the iterval I, if f is cotiuous o I ad f(a f(b for all A, B S(I satisfyig A B (i.e A B is a positive operator. The fuctio f is operator covex o I, if f is cotiuous o I ad f(sa + tb sf(a + tf(b for all A, B S(I ad for all positive umbers s ad t. The fuctio f is called operator cocave o I if f is operator covex o I. If f is a operator covex fuctio o the iterval I, T i S(I, ad w i > 0 (i =,..., such that i= w i =, the the discrete Jese s iequality is give by ( f w i T i w i f(t i. i= i= If f is a operator cocave fuctio o I, the iequality i ( is reversed. Some iterpolatios of ( are give i [7]. The power meas for strictly positive operators with positive weights are also defied i [7] ad their mootoicity is discussed. I [5], the class S(I is cosidered to give some refiemets of the discrete Jese s iequality, ad the mootoicity property of the correspodig mixed symmetric meas is studied. The iterpolatios give i [7] are special cases of some results i [5]. We start with a result from [5]. To formulate this result we eed some otatios ad some hypotheses which will also give the basic cotext of our mai results. The power set of a set X is deoted by P (X. X meas the umber of elemets i X. The usual symbol N is used for the set of atural umbers (icludig 0, while N + meas N \ {0}. (H Let I R be a iterval, ad let T i S(I ( i.

3 658 L. HORVÁTH, KHURAM ALI KHAN 2,3,, AND J. PEČARIĆ2,4 (H 2 Let w,..., w be positive umbers such that w j =. (H 3 Let the fuctio f : I R be operator covex. (H 4 Let h, g : I R be cotiuous ad strictly operator mootoe fuctios. We do ot apply Theorem. i this paper, ad therefore o the score of the exact meaig of the followig expresios A k,l (k l see [5] or [6]. Let A k,k := (i,...,i k I k ( k w is α s= Ik,i s f k w is α Ik T is,is s= k w is α Ik,is s= ad for each k l let A k,l := ( l t Ik,l (i,..., i l (k... l (i,...,i l I l s= w is α Ik,i s f Now we are i a positio to formulate oe of the mai results i [5]:, l w is α Ik T is,is s= l w is α Ik,is s=. Theorem.. Assume (H -(H 3 are satisfied. The (2 f w r T r A k,k A k,k... A k,2 A k, = r= w r f(t r. r= I this paper, we first use the method of Horváth adopted i [3] to costruct a ew refiemet of Jese s iequality for operator covex fuctios. I this way we are able to geeralize the refiemet results give i [5] as well as the results of Mod ad Pečarić i [7]. Secodly, we itroduce a parameter depedet refiemet of ( by usig the method give i [4]. With the help of this ew refiemet, we costruct the parameter depedet mixed symmetric meas for a subclass of S(I ad also give the mootoicity property of these operator meas.

4 ON PARAMETER DEPENDENT REFINEMENT OF DISCRETE JENSEN S INEQUALITY Geeralizatios To give the geeralizatio of Theorem., we start with the followig otatios itroduced i [3]: Let X be a set. For every oegative iteger m, defie P m (X := {Y X Y = m}. We itroduce two further hypotheses: (H 5 Let S,..., S be fiite, pairwise disjoit ad oempty sets, let S := S j, ad let c be a fuctio from S ito R such that c(s > 0, s S, ad c(s =, j =,...,. s S j Let the fuctio τ : S {,..., } be defied by τ(s := j, if s S j. (H 6 Suppose A P (S is a partitio of S ito pairwise disjoit ad oempty sets. Let k := max { A A A}, ad let A l := {A A A = l}, l =,..., k. (We ote that A l (l =,..., k may be the empty set, ad of course, S = k l A l. Now, we give a refiemet of (. The empty sum of umbers or vectors is take to be zero. Theorem 2.. If (H -(H 3 ad (H 5 -(H 6 are satisfied, the f w j T j N k N k... N 2 N = w j f(t j,

5 660 L. HORVÁTH, KHURAM ALI KHAN 2,3,, AND J. PEČARIĆ2,4 where N k := k ( c(sw τ(s T τ(s c(sw τ(s f s A, c(sw A Al s A τ(s s A ad for every m k the operator N k m is give by ( ( m k N k m := c(sw τ(s f(t τ(s + A A l A A l B P l m (A s A l=m+ ( m! (l... (l m ( c(sw τ(s T τ(s c(sw τ(s f s B. c(sw τ(s s B s B Proof. The proof is etirely similar to the proof of Theorem i [3], so we omit it.. The first applicatio of Theorem 2. leads to a geeralizatio of Theorem.. Theorem 2.2. Assume that (H -(H 3 are satisfied, let k be a fixed iteger, ad let I k {,..., } k. For j =,..., we cosider the sets S j := {((i,..., i k, l (i,..., i k I k, l k, i l = j}. Let c be a positive fuctio o S := S j such that c ((i,..., i k, l =, j =,...,. ((i,...,i k,l S j The (3 f w j T j N k N k... N 2 N = w j f(t j, where := (i,...,i k I k ad for every m k N k ( k k c ((i,..., i k, l w il T il c ((i,..., i k, l w il f k, c ((i,..., i k, l w il N k m := m! (k... (k m (i,...,i k I k l <...<l k m k

6 ON PARAMETER DEPENDENT REFINEMENT OF DISCRETE JENSEN S INEQUALITY ( k m c ((i,..., i k, l j w ilj f k m c ((i,..., i k, l j w ilj T ilj. c ((i,..., i k, l j w ilj k m A immediate cosequece of the previous result is Theorem.: choosig c ((i,..., i k, l = S j = α Ik,j if ((i,..., i k, l S j, it ca be checked easily that the iequality (3 correspods to the iequality (2. Theorem. has some iterestig special cases (see [5]. Theorem 2.2 geeralizes these results: apply it to either I k := {(i,..., i k {,..., } k i <... < i k }, k, or I k := {(i,..., i k {,..., } k i... i k }, k. Now we apply Theorem 2. to some special situatios which correspod to some results about operator covexity.the ext examples based o examples i [3]. Example 2.3. Let, m, r be fixed itegers, where 3, m 2 ad r 2. I this example, for every i =, 2,..., ad for every l = 0,,..., r the iteger i + l will be idetified with the uiquely determied iteger j from {,..., } for which (4 l + i j (mod. Itroducig the otatio D := {,..., } {0,..., r}, let for every j {,..., } S j := {(i, l D i + l j (mod } {j}, ad let A P (S (S := S j cotai the followig sets: A i := {(i, l D l = 0,..., r}, i =,...,

7 662 L. HORVÁTH, KHURAM ALI KHAN 2,3,, AND J. PEČARIĆ2,4 ad A := {,..., }. Let c be a positive fuctio o S such that c (i, l + c (j =, j =,...,. (i,l S j A careful verificatio shows that the sets S,..., S, the partitio A ad the fuctio c defied above satisfy the coditios (H 5 ad (H 6, τ (i, l = i + l, (i, l D, (by the agreemet (see (4, i + l is idetified with j τ (j = j, j =,...,, S j = r + 2, j =,...,, ad A i = r +, i =,...,, A =. Now we suppose (H -(H 3 are satisfied. The by Theorem 2. ( r r c (i, l w i+l T i+l f w j T j N k = c (i, l w i+l f l=0 r i= l=0 c (i, l w i+l l=0 c(jw j T j (5 + c(jw j f c(jw j I case w j f(t j. c (i, l := it follows from (5 that ( f w j :=, j =,...,, m, (i, l D, c(j := m (r + m T j m j =,...,, ( Ti + T i T i+r f r + i=

8 ON PARAMETER DEPENDENT REFINEMENT OF DISCRETE JENSEN S INEQUALITY m m f ( T j f(t j. Example 2.4. Let ad k be fixed positive itegers. Let D := {(i,..., i {,..., k} i i = + k }, ad for each j =,...,, deote S j the set S j := D {j}. For every (i,..., i D desigate by A (i,...,i the set A (i,...,i := {((i,..., i, l l =,..., }. It is obvious that S j (j =,..., ad A (i,...,i ((i,..., i D are decompositios of S := S j ito pairwise disjoit ad oempty sets, respectively. Let c be a fuctio o S such that c ((i,..., i, j > 0, ((i,..., i, j S ad (6 (i,...,i D c ((i,..., i, j =, j =,...,. I summary we have that the coditios (H 5 ad (H 6 are valid, ad τ ((i,..., i, j = j, ((i,..., i, j S. Suppose (H -(H 3 are satisfied. The by Theorem 2. f w j T j N k = ( c ((i,..., i, l w l (i,...,i D (7 f If we set c ((i,..., i, l w l T l c ((i,..., i, l w l w j f(t j. w j :=, j =,...,,

9 664 L. HORVÁTH, KHURAM ALI KHAN 2,3,, AND J. PEČARIĆ2,4 ad c ((i,..., i, j := i j ( +k k the (6 holds, sice by some combiatorial cosideratios ( + k 2 D =, ad (i,...,i D i j = + k, ( ( + k 2 + k = k I this situatio (7 ca therefore be expressed as ( ( f T j f ( +k 2 k (i,...,i D + k, j =,...,. i l T l f(t j. Let us close this sectio by derivig a sharpeed versio of the arithmetic mea - geometric mea iequality. Example 2.5. Let 2 be a fixed positive iteger, let ad let S j := { (i, j {,..., } 2 i =,..., j }, j =,...,, A i := { (i, j {,..., } 2 j = i,..., }, i =,...,. If T,..., T are strictly positive operators, the it follows from Theorem 2. that ( ( T j T T l j j=i l j i= j=i j=i j ad therefore (T... T l (T l (T, i= j=i j=i T j j j j=i j T T.

10 ON PARAMETER DEPENDENT REFINEMENT OF DISCRETE JENSEN S INEQUALITY Parameter Depedet Refiemet I this part of the paper we use the followig hypothesis: (H 7 Cosider a real umber λ such that λ. Now we give a parameter depedet refiemet of the discrete Jese s iequality (. Theorem 3.. Suppose (H -(H 3 ad (H 7. For k N, we itroduce the sets { } S k := (i,..., i N i j = k, k N, ad defie the operators (8 := The C k (λ = C k (T,..., T ; w,..., w ; λ ( w ( + λ k j f i!... i! (i,...,i S k w j T j. wj (a f w j T j = C 0 (λ C (λ... C k (λ... w j f(t j, k N. (b For every fixed λ > that lim C k(λ = k w j f(t j. It follows from the defiitio of S k that S k {0,..., k} (k N, ad it is obvious C k ( = f w j T j, k N. The proof of Theorem 3. is essetially the same as the proofs of the similar results i [4], so it is omitted. But to prove the secod part of the theorem we eed the followig two results. First, we geeralize Lemma 5 i [4].

11 666 L. HORVÁTH, KHURAM ALI KHAN 2,3,, AND J. PEČARIĆ2,4 Lemma 3.2. Let (X, be a ormed space. Let p,..., p be a discrete distributio with 2, ad let λ >. Let l {,..., } be fixed. e l deotes the vector i R that has 0s i all coordiate positios except the lth, where it has a. Let q,..., q be also a discrete distributio such that q j > 0 ( j ad q l > max (q,... q l, q l+,..., q. If { g : (t,..., t R t j > 0 ( j, is a bouded fuctio for which } t j = X τ l := lim el g exists, ad p l > 0, the lim k i!... i! qi... q i (i,...,i S k g λ i p,..., pj λ i p pj = τ l. Proof. We have to modify just the fial part of the proof of Lemma 5 i [4]. We ca suppose that l =. Choose 0 < ε <. Sice the distributio fuctio F of the Chi-square distributio (χ 2 -distributio with degrees of freedom is cotiuous, ad strictly icreasig o ]0, [, there exists a uique t ε > 0 such that F (t ε = ε. Defie Sk := (i k,..., i k S k k ( ijk k q j q j 2 < t ε, let Sk 2 := S k \ Sk (k N +, ad cosider the sequeces a k := (i k,...,i k S k i k!... i k! qi k... q i k g λ i k p,..., k pj λ i k p k pj,

12 ad ON PARAMETER DEPENDENT REFINEMENT OF DISCRETE JENSEN S INEQUALITY a 2 k := where k N +. (i k,...,i k S 2 k i k!... i k! qi k... q i k g λ i k p,..., k pj By usig the first part of the proof of Lemma 5 i [4], we have that (i (i k,...,i k S k λ i k p k pj i k!... i k! qi k... q i k = ε + δ ε (k, k N +, where lim δ ε (k = 0 (let k ε N + such that δ ε (k < ε for all k > k ε, k (ii for every ε > 0 we ca fid a iteger k ε > k ε such that for all k > k ε g λ i k p λ i k p,..., λ i jk pj λ i τ < ε, (i k,..., i k Sk. jk pj Sice g bouded o its domai ( g τ m, it follows from (i ad (ii that i!... i! qi... q i g λ i p λ i p,..., (i,...,i S k λ i j pj λ i τ j pj (i,...,i S k + (i,...,i S 2 k i!... i! qi... q i g i!... i! qi... q i g λ i p,..., pj λ i p,..., pj λ i p pj λ i p pj ε ( ε + δ ε (k + m (ε δ ε (k, k > k ε,, τ τ ad this gives the result. The secod lemma correspods to the symbolic calculus for self-adjoit operators.

13 668 L. HORVÁTH, KHURAM ALI KHAN 2,3,, AND J. PEČARIĆ2,4 Lemma 3.3. Assume (H ad let f : I R be cotiuous. Let the fuctio { } g : (t,..., t R t j > 0 ( j, t j = B(H defied by g (t,..., t := f t j T j. The lim g = f(t l, l. el Proof. Let α := mi j (mi Sp(T j ad β := max j (max Sp(T j, where Sp(T deotes the spectrum of T. The Sp t j T j [α, β] I for all t j 0 ( j with t j =. It is eough to prove that f is cotiuous o S([α, β]. To prove this let ε > 0 be fixed, ad let (A N be a sequece i S([α, β] such that A A S([α, β]. Sice f is cotiuous o [α, β], the Stoe-Weierstrass theorem implies the existece of a sequece of real polyomial fuctios (f k k N which coverges uiformly o [α, β] to f. It follows that there exists k 0 N such that f k0 (t f(t < ε, t [α, β]. 3 The fudametal result for cotiuous fuctioal calculus (see for example [2] yields that (9 f(a f k0 (A = (f f k0 (A = sup f(t f k0 (t t Sp(A sup f(t f k0 (t < ε t [α,β] 3, N, where meas the orm o H. Similarly, we have (0 f k0 (A f(a < ε 3.

14 ON PARAMETER DEPENDENT REFINEMENT OF DISCRETE JENSEN S INEQUALITY Sice A A, we obtai A i A i for every i N, ad therefore there is 0 N such that ( f k0 (A f k0 (A < ε 3 for all > 0. Now the iequalities (9- give that f(a f(a f(a f k0 (A + f k0 (A f k0 (A + f k0 (A f(a < ε for all > 0, ad hece f(a f(a. The proof is complete. Suppose (H -(H 3 ad (H 7. We cosider three special cases of (8. (a k =, N + : C (λ = (b k N, = 2 : C k (λ = + λ (λ + k (c w =... = w := : C k (λ = ( + (λ w i f i= k i=0 ( k i w j T j + (λ w i T i + (λ w i. (λ i w + λ k i w 2 f ( λ i w T + λ k i w 2 T 2 ( ( + λ k i!... i! (i,...,i S k λ i w + λ k i w 2. T j f. Next, we defie some further operator meas ad study their mootoicity ad covergece. Defiitio 3.4. We assume that (H, (H 2 ad (H 4 are satisfied ad λ. The we defie the operator meas with respect to (8 by (2 M h,g (k, λ := h ( + λ k (i,...,i S k w j i!... i!

15 670 L. HORVÁTH, KHURAM ALI KHAN 2,3,, AND J. PEČARIĆ2,4 w j g(t j (h g, k N. wj We ow give the mootoicity of the meas (2 by the virtue of Theorem 3.. Propositio 3.5. For λ, we assume (H, (H 2 ad (H 4. The (a M g = M h,g (0, λ... M h,g (k, λ... M h, k N, if either h g is operator covex ad h is operator mootoe or h g is operator cocave ad h is operator mootoe. (b M g = M h,g (0, λ... M h,g (k, λ... M h, k N, if either h g is operator covex ad h is operator mootoe or h g is operator cocave ad h is operator mootoe. (c I both cases lim M h,g(k, λ = M h. k Proof. The idea of the proof is the same as give i [5]. As a special case we cosider the followig example. Example 3.6. If I :=]0, [, h := l ad g(x := x (x ]0, [, the by Propositio 3.5 (b, we have the followig iequality: for every T j > 0 ( j, λ, ad k N + T w j j (i,...,i S k w j T j wj (+λ k λ ij w i!...i! j w j T j, which gives a sharpeed versio of the arithmetic mea - geometric mea iequality T j (i,...,i S k T j (+λ k i!...i! T j.

16 ON PARAMETER DEPENDENT REFINEMENT OF DISCRETE JENSEN S INEQUALITY Supported by the power meas we ca itroduce mixed symmetric operator meas correspodig to (8: Defiitio 3.7. Assume (H with I :=]0, [ ad (H 2. We defie the mixed symmetric meas with respect to (8 by if s, r R ad s 0. M s,r (k, λ ( := w ( + λ k j i!... i! M s r T,..., T ; (i,...,i S k λ i w,..., wj λ i w wj s, The mootoicity ad the covergece of the previous meas is studied i the ext result. Propositio 3.8. Assume (H with I :=]0, [ ad (H 2. The (a (3 M s... M s,r (k, λ... M s,r (0, λ = M r, if either (i s r or (ii r s or (iii s, r s 2r; while the reverse iequalities hold i (3 if either (iv r s or (v s r or (vi s, r s 2r. (b All of these cases for each fixed λ >. lim k M s,r(k, λ = M s

17 672 L. HORVÁTH, KHURAM ALI KHAN 2,3,, AND J. PEČARIĆ2,4 Proof. We apply Propositio 3.5 (b. Refereces [] T. Furuta, J. M. Hot, J. Pečarić ad Y. Seo, Mod-Pečarić Method i Operator Iequalities, Elemet, Zagreb (2005. [2] G. Helmberg, Itroductio to Spectral Theory i Hilbert Spaces, Joh Wiley & Sos Ic., New York, (969. [3] L. Horváth, A method to refie the discrete Jese s iequality for covex ad mid-covex fuctios, Math. Comput. Modellig 54 ( [4] L. Horváth, A parameter depedet refiemet of the discrete Jese s iequality for covex ad mid-covex fuctios, J. Iequal. Appl. 20:26, (20 4 pages. [5] L. Horváth, K. A. Kha ad J. Pečarić, Refiemets of Jese s iequality for Operator Covex Fuctios, submitted [6] L. Horváth ad J. Pečarić, A refiemet of the discrete Jese s iequality, Math. Ieq. Appl., Vol. 4, No. 4, (20, [7] B. Mod ad J. Pečarić, Remarks o Jese s Iequality for Operator Covex Fuctios, A. Uiv. Mariae Curie-Sklodowska Sec. A., 47, 0, (993,