Computers and Mathematics with Applications. On the positivity of certain trigonometric sums and their applications

Transcription

1 Computers ad Mathematics with Applicatios 6 (0) Cotets lists available at SciVerse ScieceDirect Computers ad Mathematics with Applicatios joural homepage: O the positivity of certai trigoometric sums ad their applicatios Saiful R. Modal a, A. Swamiatha b, a School of Mathematical Scieces, Uiversiti Sais Malaysia, 800 USM Peag, Malaysia b Departmet of Mathematics, Idia Istitute of Techology, Rooree , Uttarhad, Idia a r t i c l e i f o a b s t r a c t Article history: Received 4 September 00 Received i revised form 6 September 0 Accepted 9 September 0 Keywords: Positive trigoometric sum Cesáro meas Close-to-covex fuctios Starlie fuctios I this paper, we fid coditios o the coefficiets {b } such that the correspodig trigoometric (cosie ad sie) sums give respectively by b si θ > 0 ad b cos θ > 0 for all N are positive. Usig these results, we fid that the fuctios f that are i the class of aalytic fuctios A are starlie of certai order i the uit disc D by meas of coditios o the Taylor coefficiets of f. As a applicatio, we also fid coditios such that the Cesáro meas of order β of f () are close-to-covex ad starlie i D. 0 Elsevier Ltd. All rights reserved.. Itroductio Trigoometric series have bee a importat ad iterestig part of mathematics over the last ceturies ad have bee used as a importat tool i pure mathematics, particularly after Fourier series ad harmoic fuctios. Amog the cotributios made by various mathematicias such as Fejér et al., i the aspect of positivity of trigoometric sums, the most familiar oe is the Fejér Jacso Growall iequality, si θ > 0, for all N ad 0 < θ < π, (.) cojectured by Fejér i 90 ad proved idepedetly by Jacso i 9 ad by Growall i 9. Sice the, several other proofs were give ad the shortest proof is due to Ladau 3. I 953, Turá 4 established that if a si( )θ 0, 0 < θ < π, for some, the a si θ > 0, 0 < θ < π, for the same, uless all a are ero. This exhibits (.), as a cosequece of the basic iequality si( )θ 0, 0 < θ < π. Correspodig author. Tel.: addresses: saiful786@gmail.com (S.R. Modal), swamifma@iitr.eret.i, mathswami@yahoo.com, mathswami@gmail.com (A. Swamiatha) /$ see frot matter 0 Elsevier Ltd. All rights reserved. doi:0.06/j.camwa

2 387 S.R. Modal, A. Swamiatha / Computers ad Mathematics with Applicatios 6 (0) A short proof of this result is give i 5. I 997, Brow ad Wag 6 cosidered the positivity of trigoometric sums of the form T α (θ) = si θ α, 0 < θ < π ad they have show that whe is odd, T α(θ) are positive throughout the iterval 0 < θ < π wheever < α α 0 ad that α 0 =.0... is the best possible. I the case of beig eve, the positivity of T α (θ) fails to hold for all θ (0, π). I 93, aalogous to (.), Youg 7 established the iequality cos θ > 0, for all N ad 0 < θ < π. (.3) Rogosisi ad Segö 8 cosidered iequalities of the form α cos θ α > 0, for all N ad 0 < θ < π. (.4) ad observed the existece of a costat A, A ( ), such that (.4) hold for every α, < α A. I 969 Gasper 9 determied the exact value of A as follows: Lemma. (9). Let A be the positive root of the equatio 9x 7 55x 6 4x 5 948x 4 347x 3 503x 3780x 34 = 0. If < α A, the the sum T α (φ) = α cos φ cos φ cos 3φ cos φ α α 3 α α, (.5) is o-egative for 0 φ π ad for all N. However, if α > A, the T α 3 < 0 for some φ. The value of A is approximately. I 958, a surprisig ad quite deep result about the simultaeous positivity of a geeral class of cosie ad sie sum was published by Vietoris 0, which ca be stated as follows: Theorem A. Let {a } =0 be ay o-icreasig sequece of o-egative real umbers such that a 0 > 0 ad (.) a ( )a,. (.6) The for all positive itegers, we have ad a cos θ > 0, 0 θ < π, =0 a si θ > 0, 0 < θ < π. (.7) (.8) Vietoris himself observed that (.7) ad (.8) satisfy the special case a = c, where c = c = (/), = 0,,,.... (.9)! Here, by (a) we mea the Pochhammer symbol, defied by (a) 0 =, ad (a) = a(a )... (a ) = Γ ( a), =,,.... Γ (a) Iequalities (.7) ad (.8) of Vietoris exted both (.) ad (.3). The sigificace of Theorem A was uow till the appearace of the wor of Asey ad Steiig, where a simplified proof of Theorem A is give ad further show that this result has some ice applicatios i estimatig the eros of certai trigoometric polyomials. They also observed that these iequalities are better viewed i the cotext of more geeral iequalities cocerig positive sums of Jacobi polyomials ad they play a role i problems dealig with quadrature methods.

3 S.R. Modal, A. Swamiatha / Computers ad Mathematics with Applicatios 6 (0) The cosie iequality (.7) received cosiderable improvemet i the last few years. Brow ad Hewitt have show that (.7) remais true if the coditio (.6) is replaced by ( )a ()a,. I 3, Brow ad Yi gave a further geeraliatio of (.7) by showig that it remais valid uder the coditio ( β)a ( β )a,, ad β (,. (.0) They also suggested two differet (urelated) directios of possibility of additioal sharpeig of their result, by raisig the followig two questios: () Determie the maximum rage of β i (.0), for which (.7) remais true. () Modify the sequece c of (.9), by taig c = c = ( α), = 0,,,...,! ad determie the best possible rage of α for which all the cosie sums i (.7) are positive. I fact, it is expected i 3 that the upper boud for β i (.0) will be less tha.34. The complete aswers for the above two questios were give i 4. A similar result with a idepedet proof ca also be foud i 5. I 6, a systematic accout of these ew results which esure the positivity ad boudedess of partial sums of cosie or sie series were discussed. O the other had, the iequality (.8) does ot have much more improvemet. It turs out that iequality (.8) is the best possible i the sese that, if the coditio (.6) is weaeed the the correspodig sie sums are ot everywhere positive i (0, π). I a remarable result, Belov 7 obtaied a ecessary ad sufficiet coditio o the coefficiets {a } which exted both Vietoris ad Brow Hewitt s results. We state this result as a lemma because of its importace i the preset wor. Lemma.. Let a, = 0,,,... be ay decreasig sequece of positive real umbers, the the coditio ( ) a 0,, a > 0, (.) is ecessary ad sufficiet for the validity of the iequality a si θ > 0, N, 0 < θ < π. Moreover, coditio (.) implies that a cos θ > 0, N, 0 < θ < π. Note that the sequece of the coefficiets of the sums (.) does ot satisfy the coditio (.). Several applicatios that have used the geeraliatios of (.) ad (.3) are available i the literature ad have led to a deeper uderstadig of these results. Similarly, a variety of problems ca be reduced to positivity results for trigoometric or other orthogoal sums of this type. Ideed, these iequalities have remarable applicatios i the theory of Fourier series, summability theory, approximatio theory, positive quadrature methods, the theory of uivalet fuctios ad may others. We refer the reader to the recetly published research articles 8,9,4,0 ad the refereces therei for some ew results o positive trigoometric sums icludig refiemets ad extesios of (.) ad (.3) ad various applicatios. We also ote that positivity results for trigoometric sums ad geometric fuctio theory have bee closely related subjects over the past cetury. Both areas have cotributed to each other ad this paper iteds to preset few more results of this iterplay. This paper is orgaied as follows. I Sectio ey lemmas required for the wor give i the paper, mai results which are geeraliatios of previously obtaied results ad their deductios are give. I Sectio 3 applicatios of our results to the results i geometric fuctio theory (GFT) are give. I Sectio 4, usig results from earlier sectios, we fid the geometric properties of certai type of Cesáro meas ad compare with earlier ow results.. Prelimiaries ad mai results I this sectio, we deal with the partial sums of two importat trigoometric (cosie ad sie) series. Amog various results i this sectio, we geeralie the followig result give i 3.

4 3874 S.R. Modal, A. Swamiatha / Computers ad Mathematics with Applicatios 6 (0) Lemma. (3). Let α 0, b 0 =, b = ad b = followig iequalities hold: b 0 b cos φ > 0 ad b si φ > 0. Though a idepedet proof of Lemma. is give i 3, oe ca easily prove that ( ) α > 0, α 0, N = α for N,. The for all 0 < φ < π ad for all N, the ad therefore Lemma. directly follows from Lemma.. Abel s trasformatio or summatio by parts is a stadard techique i obtaiig positivity results for trigoometric sums. We state this as a lemma. Lemma.. Let {b } =0 ad {c } =0 be two sequeces of real umbers, the b c = b c j b c =0 =0 where b = b b. j=0 =0 Now we state a geeraliatio of Lemma., which ca be obtaied from a careful maipulatio of Lemma. ad the techique give i Lemma.. Theorem.. Let α 0, γ, b 0 =, b = ad b = (α) γ for N,. The for all 0 < φ < π ad for all N, the followig iequalities hold: b 0 b cos φ > 0 ad b si φ > 0. Proof. Sice b 0 b cos φ = ( α) γ ( cos φ) = ( α) (γ ) ( α) (γ ) ( α) (γ ) cos φ cos φ ( α) = cos jφ cos φ (j α) j= ad b si φ = ( α) γ si φ = ( α) (γ ) ( α) (γ ) ( α) (γ ) si φ = si φ si θ ( α) j= si jφ (j α), both are positive by the give hypothesis ad Lemma.. The followig result which is a cosequece of Theorem. ca be obtaied by applyig Lemma.. This result has may iterestig applicatios, some of them are give i Sectios 3 ad 4. Corollary.. Let α 0, γ ad a 0, a,... be a sequece of positive umbers such that, for all, ( α) γ a ( α) γ a ( α) γ a a a 0, the for all 0 < φ < π ad for all N, the followig iequalities hold: a. 0 a cos φ > 0.. a si φ > 0. Our ext result is a geeraliatio of the followig Lemma.3.

5 S.R. Modal, A. Swamiatha / Computers ad Mathematics with Applicatios 6 (0) Lemma.3 (4). For every positive iteger ad for 0 < θ < π, we have d cos θ cos θ < 0, dθ γ whe γ. This iequality fails to hold for appropriate ad θ, whe 0 < γ <. (.) Theorem.. Let α 0, γ 0. The, for every positive iteger ad for 0 < θ < π, we have d cos θ cos θ cos θ < 0. dθ ( α) γ (.) = Proof. Iequality (.) is equivalet to si(θ/) cos θ cos θ ( α) γ cos θ si θ By Theorem., si θ = cos θ = = si θ (α) γ cos θ ( α) γ > 0. = si θ ( α) γ > 0. (.3) > 0 whe α 0 ad γ. Hece the iequality (.3) will be true if we show that The left had side of the above iequality ca also be writte as cos θ cos θ ( α) = cos jθ γ ( α) γ ( cos θ) ( α) γ j = j= ( α) cos jθ γ ( α) γ > 0. j Sice cos θ > 0 by Theorem. ad (α) γ > j= (α) γ for α 0 ad γ, we have the positivity of (.3). For γ = 0, (.) is a immediate cosequece of (.). Let 0 < γ <. Write a 0 = a = ad a = (α) γ,. By Lemma. we have σ (θ) := cos θ = a cos θ where S (θ) = cos θ, S (θ) = j= Now applyig Lemma.3 we obtai d dθ σ (θ) cos θ = = (a a ) d dθ for 0 < θ < π ad the proof is complete. 3. Applicatio i GFT (a a )S (θ) a S (θ), cos jθ,. j S (θ) cos θ d a S (θ) cos θ < 0, dθ As usual, by A we mea the class of aalytic fuctios f i the uit dis D = { : < }, ormalied by the coditio f (0) = 0 = f (0) ad S = {f A : f is uivalet i D}. A fuctio f S is said to be starlie ad covex of order µ (0 µ < ), respectively, if Re f () f () > µ ad Re f () f () > µ. These classes are deoted by S (µ) ad C(µ) respectively. Note that, S (0) S ad C(0) C, respectively, are the wellow subclasses of S that map D oto domais that are starlie with respect to origi ad covex. Let T R be the subclass of S, cosistig of all typically real fuctios, i.e, all f S such that Im f ()Im () > 0. Aother importat class required for our discussio is the class of close-to-covex fuctios of order µ with respect to a fixed starlie fuctio g() ad give by the aalytic coditio Re e iη f () g() µ > 0, g S, D, (3.)

6 3876 S.R. Modal, A. Swamiatha / Computers ad Mathematics with Applicatios 6 (0) for some real η ( π/, π/). The family of all close-to-covex fuctios of order µ, relative to g S is deoted by K g (µ). If there is o specificatio about the fuctio g is give, the K g (µ) is deoted as K. Note that for 0 µ <, each fuctio i K g (µ) is uivalet i D. For a particular choice of g, we get particular classes of K g (µ). We list here oly the classes that are eeded for our results. (i) g() = K (µ) =: R(µ) = {f A : Re f () > µ} (ii) g() = ( ) K (µ) := {f A : Re (( )f ()) > µ} (iii) g() = ( ) K (µ) := {f A : Re (( )f ()) > µ} where η give i (3.) is tae as ero. Further, we have R(0) := R, K (0) := K ad K (0) := K. The followig are the ecessary ad sufficiet coditios for f to be i K. Lemma 3.. f A has real coefficiets ad f () K if, ad oly if, f () is typically real. Proof. A fuctio f A is said to be covex i the directio of the imagiary axis 5 if every lie parallel to the imagiary axis either itersects f (D) i a iterval or does ot itersect it at all. From the well-ow result give i 6, it is clear that f A has real coefficiets ad is covex i the directio of imagiary axis if, ad oly if, f () is typically real. It is also ow that f A has real coefficiets, the f is covex i the directio of imagiary axis if, ad oly if, Re (( )f ()) > 0 D which meas that f K ad the proof is complete. Remar 3.. The fuctios, ±, ±, ( ± ), (3.) ± are the oly ie fuctios which are starlie uivalet ad have iteger coefficiets i D, (see 7 for details). We ote that, it is easy to give sufficiet coditios for f to be close-to-covex, i terms of the Taylor coefficiets of f, at least whe the correspodig starlie fuctio g() taes oe of the above forms. For the iterested reader o details regardig these classes ad the correspodig results, we refer to 5,8 30. Theorem 3.. Let 0 µ < ad f A be such that f () ad f () µ f () ad Re (f () µ f () ) > 0, the f S (µ). are typically real i D. Further, if Re f () > 0 Proof. The result for µ = 0 is give i 3. It remais to prove the result for the case 0 < µ <. It is eough to prove that > 0. Cosider Re f ()/f () µ ( µ) where, ( µ)f () (f () µf ()) = g() = f () µ f (). 0 ( µ)f (t) dt = f () µ f () 0 ( µ)f (t) dt (3.3) g() Note that both f (t) ad g() are i same half plae. For, if Im > 0(< 0), the both fuctios f (t) ad g() beig typically real, their values will lie i the same plae, vi., upper half (lower half) plae. Further, as Re f () > 0 ad Re (f () µ f () ) > 0, both f (t) ad g() are also i the right half plae. Therefore, Re ( µ)f () f ()/f () µ > (f 0 Re > 0, () µf ()) ( µ) ad the proof is complete. Remar 3.. Re f () > 0 is the coditio for f () R. Further, by the defiitio of typically real fuctio we have Im f ()Im () > 0. Hece, uder the hypothesis of Theorem 3., Re ( )f () = Re ( )Re f () Im f ()Im () > 0 which implies f () K. Therefore, the result of Theorem 3. is true for f () S (µ) R K. I fact, the same remar also holds good for the followig theorem.

7 S.R. Modal, A. Swamiatha / Computers ad Mathematics with Applicatios 6 (0) Theorem 3.. Let α 0, γ, a = ad a > 0 for. If ( µ)a ( µ)a, (3 µ)a 3 ( µ)a ( α) γ ( µ)a, ad ( α)γ ( α) γ ( µ)a, for 3. The, for 0 µ <, f () = lim f () = = a is starlie of order µ, where f () is the -th partial sum of f (). Proof. Let g () = f () µ f () = ( µ) ( µ)a = b 0 b where, b 0 = ( µ) ad b = ( µ)a,. Now, by meas of a simple calculatio, we ca establish the fact that, with the give hypothesis, {b } satisfies the coditios of Theorem., which implies Re g () > 0 i D ad Im g () > 0, if Im > 0. Agai by reflectio priciple Im g () < 0, if Im < 0. Hece g () is typically real fuctio. Usig the fact, µ µ, µ <,, µ ad Theorem., oe ca easily show that, uder a give hypothesis, Re f () > 0 ad f () is typically real i D. We coclude the proof by Theorem 3. ad usig the fact that the family of starlie fuctios is ormal. Similarly by provig f () is typically real (or i other sese f () K ) ad usig Lemma 3., the followig result ca be obtaied. Theorem 3.3. Let α 0, γ, a = ad a > 0 for. If ( α) γ ( )a ( α) γ a ( α) γ a, (3.4) is true for all, the f () ad f () belogs to K, where f () = = a ad f () = lim f () = = a. Corollary 3.. Let α = 0, γ =, a = ad a > 0 for. If ( ) a a 4a, is true for all, the f () ad f () belogs to K, where f () = = a ad f () = lim f () = = a. Remar 3.3. Corollary 3. is a immediate cosequece of Theorem 3.3. But, eve by cosiderig (3.4) as a decreasig sequece, it is ot possible to get Theorem 3.3 from Corollary 3.. We support our claim by the followig example. Example 3.. let f () = = a, with ( α) γ ( α) γ a, a = ( )( α) γ a,, α > 0, γ. The by Theorem 3.3, f () belogs to K. But for all α > 0, γ, this fact caot be deduced from Corollary 3.. Because. For α > 0 ad γ =, ad, we have ( ) a a = ( )( α) a a ( α) a = ( )( α) ( α) ( α) αa = ( α) > 0.

8 3878 S.R. Modal, A. Swamiatha / Computers ad Mathematics with Applicatios 6 (0) For α = ad γ = 3, ad for, we have ( ) a a = ( )( ) 3 a ( 3) 3 a = ( )( ) 3 ( 3) 3 a which is positive for at least some. For example, tae =, Applicatio to Cesàro meas The -th Cesàro meas of order β of f () A is give by σ β (, f ) = A β A β a for all N ad β >, where A β 0 = ad Aβ I particular, we have σ β () = A β A β ( β) = A β,.. ( 3) 3 Note that σ β (, f ) = σ β () f (), where deotes the Hadamard product or covolutio, defied as (f g)() = = a b, where f () = = a ad g() = = b for D. For details about these covolutio techiques ad the correspodig properties related to the class S, we refer the iterested reader to 5,3. Amog the results available i the literature regardig the uivalece of σ β (), the followig result due to Lewis 33 is the most geeral oe. Lemma 4. (33). For β ad N we have σ β () K. By the covolutio property of covex fuctios ad close-to-covex fuctios 3, we immediately have Corollary 4.. For β, N ad f C we have σ β (, f ) K. I 34, the followig result is established which describes the covexity of Cesàro meas. Lemma 4. (34). Let β α >, f C (3 α)/. The for all N: β σ β (, f ) C (3 α)/. The correspodig result holds if C (3 α)/ is replaced by S (3 α)/ or K (3 α)/. The followig result is immediate. Corollary 4.. Let β 3, f C/S /K. The for all N: β σ β (, f ) C/S /K. Note that, β σ β (, f ) has the represetatio β σ β (, f ) = F(, ; β; ) f () where F(a, b; c; ) := F (a, b; c; ) is the well-ow Gaussia hypergeometric fuctio. We refer the iterested reader to 35,36 for bacgroud iformatio o hypergeometric fuctios. The geometric properties such as uivalecy, close-to-covexity, starlieess ad covexity of F(a, b; c; ) ad F(a, b; c; ) are iterestig questios at preset for may researchers. For example, see 8 ad refereces therei. Hece i this sectio, we are maily iterested i the followig problems. (4.) (4.)

9 S.R. Modal, A. Swamiatha / Computers ad Mathematics with Applicatios 6 (0) Problem 4.. Is it possible that f () K (S ), but β σ β (, f ) K (S )? Problem 4.. Uder what coditio (s) Corollary 4. is true for some β < 3? Theorem 4.. Let {a } be a sequece of positive real umbers such that a = ad ( β)a ( )a, N. Suppose that, for γ < ad 0 α 6 γ 4,. ( γ α)( β)a γ ( )3a 3 ad. ( α γ )( β)a ( α)( )( )a, 3. The, β σ β (, f ) is close-to-covex with respect to both the starlie fuctios ad /( ), where f () = = a. Further that, for the same coditio, β σ β (, f ) is starlie uivalet. Proof. Let g () = β σ β (, f ) = Now, for 0 r < ad 0 θ π, β = A β Re g () = b 0 r b cos θ ad Im g () = r b si θ, where, b 0 = ad for all, b = β A β A β A β a. (4.3) ( ) ( )a ( )a b = b. ( β) ( )a For a give α, a straightforward computatio gives ( α) γ = αγ (γ, )α 6 (γ ) (γ 3) ( γ )α (γ, 4)α4 3. (γ 7) ad Hece, γ = α (γ, 6)α6 45 (γ 0) αγ (γ ) 4 (6 γ )α γ α (γ, 6) 70( α) 6 γ α (γ, ) ( α) (6 γ ) 7( α),. ( α) γ b b (γ ) αγ b b = b (γ ) ( γ ) 3( α) ( αγ ) γ ( ) ( β) 3a 3 0. a (4 γ )α 0 (γ, 4) 4( α) 4 (4 γ ) 5( α) Similarly, for all 3, ( α) γ ( α) γ b b γ b b α b = ( α γ ) ( α) ( )( )a 0. α ( β)a

10 3880 S.R. Modal, A. Swamiatha / Computers ad Mathematics with Applicatios 6 (0) Therefore, {b } =0 satisfies the hypothesis of Theorem.. This clearly meas that, from the miimum priciple for harmoic fuctios, Re g () > 0. Similarly, we have either Im g () 0 i < = x i0 < or Im g () > 0 i D { : Im > 0}. The first case implies g () =, ad hece the coclusio. For the secod case, usig the reflectio priciple, we have Im g () < 0 i D { : Im < 0}. Now g () is close-to-covex with respect to, follows from Re g () > 0 i D. O the other had, Re ( )g () = Re ( )Re g () Im Im g () > 0 implies that g () is close-to-covex with respect to the starlie fuctio /( ). By virtue of Theorem 3. (with µ = 0), it is easy to coclude that g () is starlie ad this completes the proof. Example 4.. Let γ <, 0 α 6 γ ad 4 β max 0, ( ), γ αγ γ ( 3) α γ. The β σ β (, log( )) is close-to-covex with respect to ad /( ). Uder the same coditios, it is also starlie uivalet. Remar 4.. It is well ow that f () = log( ) is close-to-covex with respect to starlie fuctio /( ). Now taig α = 0, γ = i Example 4., we ca say that for 5 ad β β, where 0 β < 3, β σ β (, log( )) is close-to-covex with respect to the starlie fuctio /( ). Note that for this particular f ad 5, the same coclusio caot be obtaied by Corollary 4.. But we have o iformatio for other values of f i geeral. O the other had, for 6, the order of Cesàro mea of f () = log( ) give i Corollary 4. is better. The followig two examples will provide a partial aswer to the Problem 4.. Example 4.. For γ < ad 0 α 6 γ, let 4 β max 3( ), ( ), Cosider the fuctio γ αγ f () = log( ) 3 = =3 γ ( 3). α γ Clearly, f () = 3 ad ( )f () = 3( ). By easy computatios, we have (Re f ()) = /3 < 0, ad (Re ( )f ()) = /3 < 0.. Hece f () is ot close-to-covex with respect to the starlie fuctios ad /( ), D. I fact, f () is ot eve uivalet i D as f () = 0 at = 3 D. But, with give β, the coefficiet of f () satisfies the hypothesis of Theorem Hece β σ β (, log( ) 3 ) is close-to-covex with respect to ad /( ). It is also starlie uivalet. Example 4.3. For γ < ad 0 α 6 β, log( ) 4 σ β γ 4, is close-to-covex with respect to ad /( ) D, where γ 3( α) β max, ( ) αγ, ( 3) α γ (3 α), ( 4) 3(3 α γ ), γ ( 5). 4 α γ

11 S.R. Modal, A. Swamiatha / Computers ad Mathematics with Applicatios 6 (0) It is also starlie uivalet. Let f () = log( ) 4 = =5, D. Clearly, f () = 3 ad ( )f () = 3 ( ). By easy computatios, we have (Re f ()) = /3 < 0 ad (Re ( )f ()) = 3/4 < 0. Hece f () is ot close-to-covex with respect to both the starlie fuctios ad /( ), D. But with the give β, the coefficiet of f () satisfies the hypothesis of Theorem 4.. Hece β, log( ) 4 σ β is close-to-covex with respect to ad /( ). It is also starlie uivalet. Theorem 4.. Let {a } be a sequece of positive real umbers with a = ad satisfy the hypothesis of Theorem 4.. The σ β (, f ) R(µ ), where β µ ( )a β. Proof. Let for 0 r < ad 0 θ π, Re g () µ = b 0 µ r b cos θ, where b 0 = ad b = β ( µ ) A β µ ( )a β b 0 b. We ote that, A β ( )a,. Now ( ) ( )a b = b,. ( β) ( )a Usig this, we obtai the remaiig part of the proof, similar to the proof of Theorem 4.. We omit details. Hece, by the virtue of Theorem., we have Re g () µ µ > 0 Re g () > µ. Theorem 4.3. Let {a } be sequece of positive real umbers such that a =. If, for γ < ad 0 α 6 γ 4,. ( γ )( β)a (3 γ )( )a,. ( αγ )( 3 β)(3 γ )a γ ( 3)(4 γ )a 3, ad 3. ( α γ )( β)( γ )a ( α)( )( γ )a, 3, the, β σ β (, f ) S (γ ). Proof. where, g () = β σ β (, f ) = d, = (4.4) d = 0, ad d = β A β A β a,. It is eough to prove that uder the give hypothesis, {d } satisfies the coditios of Theorem 3.. By simple calculatio, we have ( γ )d (3 γ )d. Now ( α) γ (3 γ )d ( αγ )(3 γ ) (4 γ )d 3 γ d (4 γ )d 3 = d γ ( αγ )(3 γ ) (4 γ ) d 3 d = d γ ( αγ )(3 γ )( 3 β) (4 γ )( 3) a 3 0. a

12 388 S.R. Modal, A. Swamiatha / Computers ad Mathematics with Applicatios 6 (0) Agai, for 3, by a simple calculatio usig the give hypothesis of the theorem, we have ( γ )d ( α)γ ( α) γ ( γ )d, for 3. Hece, by Theorem 3., the result follows ad the proof is complete. Note that for γ close to, the hypothesis of the above theorem restricts the coefficiets ad hece the coefficiets {a } are too small. Numerical experimets suggest that, for γ 3/, the coefficiets {a } are comparatively bigger ad ca have further applicatios. Theorem 4.4. Let {a } be a sequece of positive real umber such that a =. Suppose that, for γ < ad 0 α 6 γ, 4. ( β)a γ ( )a,. ( α γ )( β)a ( α)( )( )a,. The, β σ β (, f ) is close-to-covex with respect to the starlie fuctio. Proof. Cosider g () give i (4.3). The for 0 r < ad 0 θ π, we have Im g () = r b si θ (4.5) where, Now, b = β A β A β a b = β ( α) γ b b (γ ) αγ b b = b ( )a a b ;. (γ ) = b (γ ) ( αγ ) (γ ) b b ( αγ ) (γ ) ( ) ( β) Similarly, for, ( α) γ ( α) γ b b γ b b = b ( α γ ) ( α) b α α b = b ( ) ( )a ( α γ ) ( α), α ( β) a a 0. which is o-egative. Now by the same argumet as Theorem 4., g () is typically real i D. Hece by Lemma 3., we have the result. β Remar 4.. Sice, for β = 0, σ β (, f ) = f (), ad as the class of all close-to-covex fuctios with respect to a particular starlie fuctio is a Normal family, f () = lim f () is also close-to-covex with respect to the same starlie fuctio. By the same argumet f () is also starlie whe f () is starlie. Note that, with referece to Remar 3., we have o result for the close-to-covexity of β σ β (, f ) with respect to the starlie fuctios /( ) ad /( ). Hece it will be iterestig if oe ca fid results i this directio. I particular, with respect to the starlie fuctio /( ), there are ot may results o close-to-covexity of fuctios f A i the literature. a Acowledgmet The authors wish to tha the aoymous referee for the suggestios which led to a improvemet of the paper.

13 S.R. Modal, A. Swamiatha / Computers ad Mathematics with Applicatios 6 (0) Refereces D. Jacso, Über eie trigoometrische Summe, Red. Circ. Mat. Palermo 3 (9) T.H. Growall, Über die Gibbssche Erscheiug ud die trigoometrische Summe si x / si x / si x, Math. A. 7 () (9) E. Ladau, Über eie trigoometrische Ugleichug, Math. Z. 37 () (933) P. Turá, O a trigoometrical sum, A. Soc. Polo. Math. 5 (953) R. Asey, J. Fitch, G. Gasper, O a positive trigoometric sum, Proc. Amer. Math. Soc. 9 (968) G. Brow, K.-Y. Wag, A extesio of the Fejér Jacso iequality, J. Aust. Math. Soc. Ser. A 6 () (997). 7 W.H. Youg, O certai series of fourier, Proc. Lod. Math. Soc. (93) W. Rogosisi, G. Segö, Über die Abschitte vo Potereihe, die i eiem Kreise beschrät bleibe, Math. Z. 8 () (98) G. Gasper, Noegative sums of cosie, ultraspherical ad Jacobi polyomials, J. Math. Aal. Appl. 6 (969) L. Vietoris, Über das Voreiche gewisser trigometrishcher Summe, Situgsber, Oest. Aad. Wiss. 67 (958) R. Asey, J. Steiig, Some positive trigoometric sums, Tras. Amer. Math. Soc. 87 (974) G. Brow, E. Hewitt, A class of positive trigoometric sums, Math. A. 68 () (984) 9. 3 G. Brow, Q. Yi, Positivity of a class of cosie sums, Acta Sci. Math. (Seged) 67 ( ) (00) S. Koumados, A extesio of Vietoris iequalities, Ramauja J. 4 () (007) G. Brow, F. Dai, K. Wag, Extesios of Vietoris s iequalities. I, Ramauja J. 4 (3) (007) G. Brow, Positivity ad boudedess of trigoometric sums, Aal. Theory Appl. 3 (4) (007) A.S. Belov, Mat. Sb. 86 (4) (995) 46; traslatio i Sb. Math. 86 (4) (995) H. Aler, S. Koumados, A ew refiemet of Youg s iequality, Proc. Edib. Math. Soc. () 50 () (007) S. Koumados, Positive trigoometric sums ad applicatios, A. of Math. Iform. 33 (006) S. Koumados, S. Ruscheweyh, Positive Gegebauer polyomial sums ad applicatios to starlie fuctios, Costr. Approx. 3 () (006) S. Koumados, S. Ruscheweyh, O a cojecture for trigoometric sums ad starlie fuctios, J. Approx. Theory 49 () (007) G. Brow, K.Y. Wag, D.C. Wilso, Positivity of some basic cosie sums, Math. Proc. Cambridge Philos. Soc. 4 (3) (993) A.P. Acharya, Uivalece criteria for aalytic fuctios ad applicatios to hypergeometric fuctios, Ph.D. Thesis, Uiversity of Würburg, G. Brow, S. Koumados, O a mootoic trigoometric sum, Moatsh. Math. 3 () (997) P.L. Dure, Uivalet fuctios, i: Grudlehre der Mathematische Wisseschafte, vol. 59, Spriger, New Yor, M.S. Robertso, Power series with multiply mootoic coefficiets, Michiga Math. J. 6 (969) B. Friedma, Two theorems o schlicht fuctios, Due Math. J. 3 (946) R. Küster, O the order of starlieess of the shifted Gauss hypergeometric fuctio, J. Math. Aal. Appl. 334 () (007) M.S. Robertso, O the theory of uivalet fuctios, A. of Math. () 37 () (936) S. Ruscheweyh, New criteria for uivalet fuctios, Proc. Amer. Math. Soc. 49 (975) S. Ruscheweyh, Coefficiet coditios for starlie fuctios, Glasgow Math. J. 9 () (987) S. Ruscheweyh, Covolutios i Geometric Fuctio Theory, i: Sémiaire de Mathématiques Supérieures, vol. 83, Presses Uiv. Motréal, Motreal, QC, J.L. Lewis, Applicatios of a covolutio theorem to Jacobi polyomials, SIAM J. Math. Aal. 0 (6) (979) S. Ruscheweyh, Geometric properties of the Cesàro meas, Results Math. (3 4) (99) G.E. Adrews, R. Asey, R. Roy, Special fuctios, i: Ecyclopedia of Mathematics ad its Applicatios, vol. 7, Cambridge Uiv. Press, Cambridge, N.M. Temme, Special Fuctios, i: A Wiley-Itersciece Publicatio, Wiley, New Yor, 996.