Global Journal of Pure and Applied Mathematics. ISSN 0973-768 Volume 3, Number 7 (07), pp. 3467-3475 Research India Publications http://www.ripublication.com/gjpam.htm Second Hankel determinant obtained with New Integral Operator defined by Polylogarithm Function S. M. Patil Department of Applied Sciences, SSVPS B.S. Deore College of Engineering, Deopur, Dhule, Maharashtra, India. S. M. Khairnar Professor & Head, Department of Engineering Sciences, MIT Academy of Engineering, Alandi, Pune-405, Maharashtra, INDIA. Abstract By using polylogarithm function, a new integral operator is introduced. By using this operator a new subclass of analytic functions are introduced for these classes we obtained sharp upper bounds for functional a a 4 a 3. AMS subject classification: Keywords: Integral Operator, Analytic Function, Hankel Determinant, Polylogarithm Function.. Introduction In 966, Pommerenke stated the q th Hankel determinant for q,&n 0as a n a n+ a n+q+ H q (n) = a n+.. a n+q. a n+q (.) where a n s are the coefficients of various power of z in f(z).
S. M. Patil and S. M. Khairnar This determinant has also been considered by several authors. For example Noor determined the rate of growth of H q (n) an n for function f, with bounded boundary. One can easily observe that Fekete and Szegö functional H (). Fekete and Szegö then further generalized and estimate a 3 µa where µ is real & f S. We consider the Hankel determinant for the case q=andn=, H () = a a 3 a 3 a 4 = a a 4 a3 (.) We recall here the definition of well known generalization of the polylogarithm function φ(a,z) given by φ(a,z) = k= z k k a (a N,z E) (.3) Let φ δ (a, z) denote the well known generalization of the Riemann Zeta & polylogarithm function or simply the δ th order polylogarithm function given by, φ δ (a, z) = k= z k (k + a) δ (.4) where any term with k+a=0isexcluded. Using the definition of the Gamma function, a simply transformation produces the integral formula, φ δ (a, z) = Ɣ(δ) 0 ( z log ) δ t a δt Ra >, Rδ > (.5) t tz z Note that φ (0,z)= is koebe function for more details about polylogarithms ( z) in theory of univalent functions see Punnusamy & Sabapathy. Recently Khalifa Alshaqsi introduced a certain Integral Operator Ia δ defined by, Ia δ ( + a)δ f(z)= Ɣ(δ) 0 t a ( log t ) δ f (tz)δt a > 0,δ >.z E (.6) We also note that the operator Ia δ f(z) defined by [] can be expressed by the series expansion as follows, Ia δ f(z)= z + ( ) δ + a a k z k (.7) k + a obviously, we have for (δ, λ 0) k= Ia δ (I a λ δ+λ f(z))= Ia f(z) (.8)
Second Hankel Determinant with Polylogarithm Function 3 and Moreover from (.7) if follows that, We note that, I δ a (zf (z)) = z(i δ a f(z)) (.9) z(ia δ+ f(z) ) = ()Ia δ f(z) aiδ+ a f(z) (.0) Fora=0andδ = n (n is any integer) the multiplier transformation I n 0 f(z)= I n f(z) was studied by Flett and Salagean. Fora=0andδ =-n(n N 0 {0,,, 3,...}), the differential operator I n 0 f(z) = D n f(z)was studied by Salagen. Fora=andδ = n (n is any integer) the operator I n f(z) = I n f(z)was studied by Uralegaddi and Somanatha. Fora=,themultiplier transformation I f f(z) = I δ f(z) was studied by Jung et al. Fora=k-(k>), the integral operator Ik δ f(z), I k δ f(z) was studied by Komatu using the operator Ia δ, we now introduced the following cases. Definition.. We say that a function f A is in the class S aδ (b) if ) { R + (z (I δa f(z) )} b Ia δf(z) > 0 (a > 0,δ 0,b c\{0}; z E) (.) Definition.. We say that a function f A is in class C aδ (b) if ) { R + z (I δa f(z) } ) p > 0 (a > 0,δ 0,b c\{0}; z E) (.) (I δa f(z) Note that f C aδ (b) zf S aδ (b) (.3) In particular, we have starlike & convex function classes S a0 ()=S and C a0 ()=C respective. Lemma.3. Let p P then c k, k =,,...and the inequality is sharp.
4 S. M. Patil and S. M. Khairnar Lemma.4. Let p P then c = c + x(4 c ) 4c 3 = c 3 + xc (4 c ) x c (4 c ) + y( x )(4 c ) (.4) for some x and y such that x, y. Theorem.5. If f S aδ (b) then a a 4 a 3 b ( a + 3 Proof. By the definition of the class S aδ (b), there exist p P such that, + ( z(i δ a f(z)) ) p Ia δf(z) = p(z) z(i δ a f(z)) I δ a f(z) Let I δ a = z + A z + a 3 z +..., where, ( ) δ A = a a + ( ) δ A 3 = a 3 a + ( ) δ A 4 = a 4 a + so that, z{ + A z + 3A 3 z + 4A 3 z 3 +...} z + A z + A 3 z 3 +... Simplify and equating the coefficient of z on both side Equating the coefficient of z 3 on both side ) δ = b + bp(z) (.5) = b + b[ + c z + c z + c 3 z 3 ] (.6) A = A A b + A b + bc A = bc ( ) δ (.7) a + A = bc A 3 = b [c + bc ] [ ( ) b a + δ ] (.8) a 3 = (c + bc )
Second Hankel Determinant with Polylogarithm Function 5 Equating the coefficients of z 4 It is establish that, 3A 4 = b6c c + b c 3 b c c + bc 3 A 4 = 3b c c + b c 3 + bc 3 6 ( ) a + 4 δ (3b c c + b c 3 a 4 = + bc 3) 6 (.9) ( a + a a 4 a3 = (( a + 3 By using Lemma, ) δ ( ) a + 4 δ (3b c c + b c 4 + bc c 3 ) 6 δ ) b (c + bc ) ) (.0) c = c + x(4 c ) for some x 4c 3 = c 3 + c (4 c )x c (4 c )x + (4 c )( x )z for some real value of zwith z (.) ( ) a + δ ( ) a + 4 δ [ ( a a 4 a3 = c 3b c + x(4 c ) ) ] + b c 4 6 + bc [ c 3 + c (4 c )x c (4 c )x + (4 c )( ] x z) 4 {( ) a + 3 δ [ b c + x(4 c ) + bc ]} (.) Since c,c = c assume without restriction, c [0, ], we obtain by using triangle inequality x = ρ. c = c. [{( a + a a 4 a3 ) δ ( a + 4 ) δ 6 [ c 3 + c(4 c )ρ ρ (4 c )ρ F(ρ) [ 3b c 4 + 3b c (4 c ] )ρ + b c 4 + b c ] ( ) a + 3 δ b [ ρ(4 c ] ) }] + 4 (.3)
6 S. M. Patil and S. M. Khairnar {( ) a + δ ( ) a + 4 δ ( F 3b c (4 c ) ) (ρ) = 6 ( ) a + 3 δ b [ ρ(4 c ] ) )} + 4 + bc [ ] c(4 c ) ρ(4 c )ρ (.4) F (ρ) > 0 for ρ>0, implies that F is an increasing function. The upper bound for (5) are c = 0. ( ) a + 3 δ a a 4 a3 b Corollary.6. If we put δ = 0, b = we get, S a0 () = S a a 4 a 3 this result is coincide the result with Janteng. Corollary.7. If f c aδ (b) then, + ( z(i δ a f(z) ) ) b (Ia δ = p(z) f(z) ) z(i a δf(z) ) (Ia δf(z) ) = bp(z) b (.5) Simplify & equating the coefficients we get, a = bc ( ) a + δ ( ) a + 3 δ [ b c a 3 = + bc ] 6 ( ) a + 4 δ [ 3b c c + b c 3 a 4 = + bc ] 3 4 (.6) and calculate the same a a 4 a3 then we get the result of convex function. After simplification we put δ = 0, b = then we get a a 4 a3 results is coincide the 8 result with Janteng.
Second Hankel Determinant with Polylogarithm Function 7 References [] C.I. Gerhardt, G.W. Leibniz, Mathematische Schriften III/, George Olms Verlag, Hildesheimand New York, 97, 336 339. [] L. Lewin, Polylogarithms and associated functions, Elsevier North-Holland, New York and Oxford, 98. [3] A.B. Goncharov, Polylogarithms in arithmatic and geometry, Proceedings of the international Congress of Mathemticians, Zrich, Switzerland 994, 374 387. [4] L. Lewin, Strucutural properties of polylogarithms, Math, Survey and Monographs, Amer. Math. Soc., Providence, RI, 37, 99. [5] Shu Oi, Gauss hypergeometric functions, multiple polylogarithms and multiple zeta values, Publ. RIMS, Kyoto Univ., 45, 009, 98 009 (Research institute for Mathematical Sciences, Kyoto University). [6] S. Ponnusamy, S. Sabapathy, Polylogarithms in the theory of univalent functions, Results in Mathematics, 30, 996, Issue -, 36 50. [7] K. Al-shaqsi, M. Darus, An operator defined by convolution involving the polylogarithms functions, Journal of Mathematics and Statistics, 4, 008, 46 50. [8] K. Al-shaqsi, M. Darus, A multiplier transformation defined by convolution involving n th order polylogarithm functions, International Mathematical Forum, 4, 009, No. 37, 83 837. [9] J.L. Liu, H.M. Srivastava, Classes of meromorphically multivalent functions associated with the generalized hypergeometric function, Mathematical and Computer Modelling, 39, 004, 34. [0] J.E. Littlewood, On inequalities in the theory of functions, Proc. London Math Soc., 3. [] P. L. Duren, Univalent Functions, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 983. [] R. Ehrenborg, The Hankel determinant of exponantial polynomials, American Mathematical Monthly, 07 (000), 557 560. [3] M. Fekete and G. Szego, Eine Bemerkung uber ungerade schlichte Funktionen, J. London Math. Soc, 8 (933), 85 89. [4] U. Grenander and G. Szego, Toeplitz forms and their application, Univ. of Calofornia Press,Berkely and Los Angeles, (958). [5] T. Hayami and S. Owa, Hankel determinant for p-valently starlike and convex functions of order α, General Math., 7 (009), 9 44. [6] T. Hayami and S. Owa, Generalized Hankel determinant for certain classes, Int. J. Math. Anal., 4 (00), 573 585.
8 S. M. Patil and S. M. Khairnar [7] A. Janteng, S. A. Halim, and M. Darus, Coeficient inequality for a function whose derivative has positive real part, J. Ineq. Pure and Appl. Math, 7(), (006), 5. [8] A. Janteng, Halim, S. A. and Darus M., Hankel Determinant For Starlike and Convex Functions, Int. Journal of Math. Analysis, I (3) (007), 69 65. [9] J. W. Layman, The Hankel transform and some of its properties, J. of integer sequences, 4 (00),. [0] R.J. Libera, and E.J. Zlotkiewicz, Early coeficients of the inverse of a regular convex function, Proc. Amer. Math. Soc., 85(), (98), 5 30. [] R.J. Libera, and E.J. Zlotkiewicz, Coeficient bounds for the inverse of a function with derivative in P, Proc. Amer. Math. Soc., 87() (983), 5 89. [] G. Murugusundaramoorthy and N. Magesh, Coeficient Inequalities For Certain Classes of Analytic Functions Associated with Hankel Determinant, Bulletin of Math. Anal. Appl., I (3) (009), 85 89. [3] J. W. Noonan and D. K. Thomas, On the second Hankel Determinant of a really mean p valent functions, Trans. Amer. Math. Soc, 3(), (976), 337 346. [4] K. I. Noor, Hankel determinant problem for the class of functions with bounded boundary rotation, Rev. Roum. Math. Pures Et Appl, 8(8), (983), 73 739. [5] S. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc. 49 (975) 09 5. [6] S. Goyal, R. K. Laddha, On the generalized Riemann zeta function and the generalized Lambert transform, Ganita Sandesh, (997), 99 08. [7] S. C. Soh and D. Mohamad, Coeficient Bounds For Certain Classes of Close-to- Convex Functions, Int. Journal of Math. Analysis, (7) (008), 343 35. [8] T. Yavuz, Second hankel determinant problem for a certain subclass of univalent functions, International Journal of Mathematical Analysis Vol. 9(0), (05), 493 498. [9] Altinkaya Ş, Yalçin S. Construction of second Hankel determinant for a new subclass of bi-univalent functions. Turk J Math; doi: 0.3906/mat-507-39. [30] Altinkaya Ş, Yalçin S. Faber polynomial coe cient bounds for a subclass of biunivalent functions. C R Acad Sci Paris Sér I 05; 353: 075 080. [3] Brannan DA, Clunie JG. Aspects of contemporary complex analysis., Proceedings of the NATO Advanced Study Institute held at the University of Durham, Durham, July 0, 979, Academic Press, New York, London, 980. [3] Brannan DA, Taha TS. On some classes of bi-univalent functions. in: S. M. Mazhar, A. Hamoui and N. S. Faour (Eds.), Math. Anal. and Appl., Kuwait; February 8., 985, in: KFAS Proceedings Series, vol. 3, Pergamon Press, Elsevier Science Limited, Oxford, 988, pp. 53 60. see also Studia Univ. Babeş-Bolyai Math. 986; 3: 70 77.
Second Hankel Determinant with Polylogarithm Function 9 [33] Bulut S. Faber polynomial coefficient estimates for a comprehensive subclass of analytic bi-univalent functions. C R Acad Sci Paris Sér I 04; 35: 479 484. [34] Cantor DG. Power series with integral coefficients. Bull Amer Math Soc 963; 69: 36 366. [35] Çağlar M, Orhan H, Yağmur N. Coefficient bounds for new subclasses of biunivalent functions. Filomat 03; 7: 65 7.