Convolutions of harmonic right half-plane mappings
|
|
- Derek Mathews
- 6 years ago
- Views:
Transcription
1 Open Math. 06; Open Mathematics Open Access Research Article YingChun Li and ZhiHong Liu* Convolutions of harmonic right half-plane mappings OI 0.55/math Received March, 06; accepted July 7, 06. Abstract We first prove that the convolution of a normalied right half-plane mapping with another subclass of normalied right half-plane mappings with the dilatation.a C /=. C a/ is CH (convex in the horiontal direction) provided a or a 0. Secondly, we give a simply method to prove the convolution of two special subclasses of harmonic univalent mappings in the right half-plane is CH which was proved by Kumar et al. [, Theorem.]. In addition, we derive the convolution of harmonic univalent mappings involving the generalied harmonic right half-plane mappings is CH. Finally, we present two examples of harmonic mappings to illuminate our main results. Keywords Harmonic univalent mappings, Harmonic convolution, Generalied right half-plane mappings MSC 30C45, 58E0 Introduction and main results Assume that f uciv is a complex-valued harmonic function defined on the open unit disk U f C W jj < g, where u and v are real harmonic functions in U. Such function can be expressed as f h C g, where X X h./ C a n n and g./ b n n n are analytic in U, h and g are known as the analytic part and co-analytic part of f, respectively. A harmonic mapping f hcg is locally univalent and sense-preserving if and only if the dilatation of f defined by!./ g 0./=h 0./, satisfies j!./j < for all U. We denote by S H the class of all harmonic, sense-preserving and univalent mappings f h C g in U, which are normalied by the conditions h.0/ g.0/ 0 and h 0.0/. Let SH 0 be the subset of all f S H in which g 0.0/ 0. Further, let K H ; C H (resp. KH 0 ; C0 H ) be the subclass of S H (resp. SH 0 ) whose images are convex and close-to-convex domains. A domain is said to be convex in the horiontal direction (CH) if every line parallel to the real axis has a connected intersection with. For harmonic univalent functions X X f./ h./ C g./ C a n n C b n n and n n n n X X F./ H./ C G./ C A n n C B n n ; n YingChun Li College of Mathematics, Honghe University, Mengi 6699, Yunnan, China, liyingchunmath@63.com *Corresponding Author ZhiHong Liu College of Mathematics, Honghe University, Mengi 6699, Yunnan, China and School of Mathematics and Econometrics, Hunan University, Changsha 4008, Hunan, China, liuhihongmath@63.com 06 Li and Liu, published by e Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-Noerivs 3.0 License. ownload ate //8 84 AM
2 790 Y. Li, Z. Liu the convolution (or Hadamard product) of them is given by X X.f F /./.h H /./ C.g G/./ C a n A n n C b n B n n Many research papers in recent years have studied the convolution or Hadamard product of planar harmonic mappings, see [ ]. However, corresponding questions for the class of univalent harmonic mappings seem to be difficult to handle as can be seen from the recent investigations of the authors [3 7]. In [], Kumar et al. constructed the harmonic functions f a h a C g a K H in the right half-plane, which satisfy the conditions h a C g a =. / with! a./.a /=. a/. < a < ). By using the technique of shear construction (see [8]), we have h a./ n Ca. / and g a./ n a Ca (). / Obviously, for a 0, f 0./ h 0./ C g 0./ KH 0 is the standard right half-plane mapping, where h 0./ and g. / 0./. / () Recently orff et al. studied the convolution of harmonic univalent mappings in the right half-plane (cf. [4, 5]). They proved that Theorem A ([4, Theorem 5]). Let f h C g ; f h C g SH 0 with h i C g i =. / for i ;. If f f is locally univalent and sense-preserving, then f f SH 0 and convex in the horiontal direction. Theorem B ([5, Theorem 3]). Let f n h C g SH 0 with h C g =. / and!./ g0./=h 0./ e i n. R; n N C /. If n ;, then f 0 f n SH 0 and is convex in the horiontal direction. Theorem C ([5, Theorem 4]). Let f h C g KH 0 with h./ C g./ =. with a. ; /. Then f 0 f SH 0 and is convex in the horiontal direction. / and!./. C a/=. C a/ We now begin to state the elementary result concerning the convolutions of f 0 with the other special subclass harmonic mappings. Theorem.. Let f h C g SH 0 with h./ C g./ =. / and!./. C a/=. C a/, then f 0 f SH 0 and is convex in the horiontal direction for a or a 0. The following generalied right half-plane harmonic univalent mappings were introduced by Muir [7] L c./ H c./ C G c./ C c C c. / C C c c. /. UI c > 0/ (3) Clearly, L./ f 0./, it was proved in [7] that L c.u/ fre.!/ > f h C g S H, then the above representation gives that =. C c/g for each c > 0. Moreover, if L c f h C ch0 C c The following Cohn s Rule is helpful in proving our main results. Cohn s Rule ([9, p.375]). Given a polynomial of degree n, let C g cg0 C c (4) p./ p 0./ a n;0 n C a n ;0 n C C a ;0 C a 0;0.a n;0 0/ (5) p./ p 0./ n p.=/ a n;0 C a n ;0 C C a ;0 n C a 0;0 n (6) ownload ate //8 84 AM
3 Convolutions of harmonic right half-plane mappings 79 enote by r and s the number of eros of p./ inside the unit circle and on it, respectively. If ja 0;0 j < ja n;0 j, then p./ a n;0p./ a 0;0 p./ is of degree n with r r and s s the number of eros of p./ inside the unit circle and on it, respectively. It should be remarked that Cohn s Rule and Schur-Cohn s algorithm [9, p. 378] are important tools to prove harmonic mappings are locally univalent and sense-preserving. Some related works have been done on these topics, one can refer to [, 3, 6, 3, 5, 6]. In [], the authors proved the following result. Theorem. Let f a h a C g a be given by (). If f n h C g is the right half-plane mapping given by h C g =. / with!./ e i n. R; n N C /, then f a f n S H is CH for a Œ n nc ; /. In this paper, we will use a new method to prove the above theorem. The main difference of our work from [] is that we construct a sequence of functions for finding all eros of polynomials which are in U, and we will show that the dilatation of f a f n satisfies je!./j j.g a g/ 0 =.h a h/ 0 j < by using mathematical induction, which greatly simplifies the calculation compared with the proof of Theorem. in []. We also show that L c f a is univalent and convex in the horiontal direction for 0 < c. C a/=. a/, and derive the following theorem. Theorem.. Let L c H c C G c be harmonic mappings given by (3). If f a h a C g a is the right half-plane mappings given by (), then L c f a is univalent and convex in the horiontal direction for 0 < c.ca/=. a/. Recently, Liu and Li [6] defined a subclass of harmonic mappings defined by P c./ H c./ G c./ c C c. / C C c C c. / They proved the following result.. UI c > 0/ (7) Theorem E. ([6, Theorem 7]) Let P c./ be harmonic mappings defined by (7) and f n hcg with h g =. / and dilatation!./ e i n. R; n N C /. Then P c f n is univalent and convex in the horiontal direction for 0 < c =n. Similar to the approach used in the proof of Theorem E, we get the following result. Theorem.3. Let L c H c C G c be harmonic mappings given by (3) and f n h C g with h C g =. / and dilatation!./ e i n. R; n N C /. Then L c f n is univalent and convex in the horiontal direction for 0 < c =n. Remark.4. In Theorem. and Theorem.3, let c, clearly, L f 0, so Theorem. and Theorem.3 are generaliation of Theorem C and Theorem B, respectively. It also explains why Theorem B does not hold for n 3, since c, according to Theorem.3, it follows that 0 < c =n hold for n ;. Preliminary results In this section, we will give the following lemmas which play an important role in proving the main results. Lemma. ([5, Eq.(6)]). If f h C g SH 0 with h C g =. / the dilatation of f 0 f is given by and dilatation!./ g0./=h 0./, then e!./! C! C!!0 C!0!0 C!0 (8) ownload ate //8 84 AM
4 79 Y. Li, Z. Liu Lemma. ([, Lemma.]). Let f a h a C g a be defined by () and f h C g SH 0 be right half-plane mapping, where h C g =. / with dilatation!./ g 0./=h 0./.h 0./ 0; U/. Then e!, the dilatation of f a f, is given by e!./.a /!. C!/ C.a /!0. /. a/. C!/ C.a /! 0. / (9) Lemma.3. Let L c H c C G c be defined by (3) and f h C g SH 0 be right half-plane mapping, where h C g =. / with dilatation!./ g 0./=h 0./.h 0./ 0; U/. Then e!, the dilatation of L c f, is given by e!./ Œ. c/. C c/!. C!/ c!0. / Œ. C c/. c/. C!/ c! 0. / (0) Proof. By (4), we know that e!./.g cg0 / 0.h C ch 0 / 0 Similar calculation as in the proof of [6, Lemma 7] gives (0). Lemma.4 ([0, Theorem 5.3]). A harmonic function f h C g locally univalent in U is a univalent mapping of U onto a domain convex in the horiontal direction if and only if h g is a conformal univalent mapping of U onto a domain convex in the horiontal direction. Lemma.5 (See []). Let f be an analytic function in U with f.0/ 0 and f 0.0/ 0, and let where ; R. If Then f is convex in the horiontal direction. './. C e i /. C e i / ; () f 0./ Re > 0. U/ './ Lemma.6. Let L c H c C G c be a mapping given by (3) and f h C g be the right half-plane mapping with h C g =. /. If L c f is locally univalent, then L c f SH 0 and is convex in the horiontal direction. Proof. Recalling that L c H c C G c and Hence H c C G c. C c/. / ; h C g h g C c.h c C G c /.h g/ C c.h c h H c g C G c h G c g/; H c G c.h c G c /.h C g/.h c h C H c g G c h G c g/ Thus Next, we will show that S, we have H c h G c g Re ' C c.h0 g 0 / C.Hc 0 Gc 0 / C c.h g/ C.H c G c / () Cc.h g/c.h c G c / is convex in the horiontal direction. Letting './ =. / 8 < Re 8 < Re h Cc.h0 C g 0 / h 0 g 0 h 0 Cg 0 h Cc.h0 C g 0 /! C! C.H 0 c C G0 c / H 0 c G 0 c H 0 ccg 0 c ' C.Hc 0 C G0 c /!c ' C! c i i 9 = 9 = ; ; ownload ate //8 84 AM
5 Convolutions of harmonic right half-plane mappings 793 ( ) C c Re Œr./ C r. / c./. / C c Re fr./ C r c./g > 0; where r./!./ C!./ ; r c./! c./ C! c./. Therefore, by Lemma.5 and the equation (), we know that H c h G c g is convex in the horiontal direction. Finally, since we assumed that L c f is locally univalent and H c h G c g is convex in the horiontal direction, we apply Lemma.4 to obtain that L c f H c h C G c g is convex in the horiontal direction. 3 Proofs of theorems Proof of Theorem.. By Theorem A, we know that f 0 f is convex in the horiontal direction. Now we need to establish that f 0 f is locally univalent. Substituting!./. C a/=. C a/ into (8) and simplifying, yields e!./ 3 C C3a C. C a/ C a q./ C C3a C. C a/ C a 3 q./. A/. B/. C /. A/. B/. C / ; (3) If a, then q./ 3 C 5 C C. C /. C / has all its three eros in U. By Cohn s Rule, so je!./j < for all U. If a 0, it is clearly that je!./j j j < for all U. If a < 0, we apply Cohn s Rule to q./ 3 C C3a C. C a/ C a. Note that j a j <, thus we get Since Q./ a 3q./ a 0 q./ 4 4 a ˇˇˇ 4 C a 3a ˇ 4 a ˇ ˇ ˇ we use Cohn s Rule on q./ again, we get q./ q./ C. C a a / 4 a C C a. a/ 4 a 4 C a 3a 4 a ˇ <.as a < 0/;! W 4 4Ca 3a q 4 a./! 4a.a /.4 C a a / C a a C.4 a / 4 C a a 4 a q./ Clearly, q./ has one ero at We show that j 0 j, or equivalently, 0 C a a 4 C a a j C a a j j4 C a a j. C a a /.4 C a a / 3.4 a /.a / 0 Therefore, by Cohn s Rule, q./ has all its three eros in U, that is A; B; C U and so je!./j < for all U. A new proof of Theorem. By Theorem A, it suffices to show that the dilatation of f a f n satisfies je!./j < for all U. Setting!./ e i n in (9), we have " nc e!./ n e i a n C. C an n/e # i C i.n a an/e a C. C an n/ei n C.n a n i p./ e an/ei nc p./ ; (4) where and p./ nc a n C. C an n/e i C.n a an/e i ; (5) p./ nc p.=/ ownload ate //8 84 AM
6 794 Y. Li, Z. Liu Firstly, we will show that je!./j < for a n n. In this case, substituting a into (4), we have nc nc ˇ nc n n nc e i C e i ˇ je!./j ˇ ˇ i nc n n e n nc n nc ei n C e i nc ˇ n e i ei nc n nc ei n n nc C n nc n nc ei n C e i nc ˇ j n e i j < Next, we will show that je!./j < for n nc < a <. Obviously, if 0 is a ero of p./, then = 0 is a ero of p./. Hence, if A ; A ; ; A nc are the eros of p./ (not necessarily distinct), then we can write Now for ja j j, A j A j e!./ n e i. A /. A /. A /. A /. A nc /. A nc /.j ; ; ; n C / maps U onto U. It suffices to show that all eros of (5) lie on U for n nc < a <. Since ja 0;0j j.n a an/e i j < ja nc;0 j, then by using Cohn s Rule on p./, we get p./ a nc;0p./ a 0;0 p./. a/. C n/œ. C a/. a/n 4 p./.n a an/e i p./ n n n C n C n C e i Since n. a/.cn/œ.ca/. a/n < a <, we have > 0. Let q nc 4./ n n ja 0; j j nc e i j < ja n; j, by using Cohn s Rule on q./ again, we obtain p./ a n;q./ a 0; q./ q./ n.n C 4/.n C / nc e i q./ n n C n C 4 n C n C 4 e i nc n C nc e i, since Let q./ n nc nc4 n C nc4 e i, then ja 0; j nc4 < ja n ;j, we get p 3./ a n ;q./ a 0; q./ q./.n C /.n C 6/.n C 4/ nc4 e i q./ n n C 4 n C 6 n 3 C n C 6 e i Let q 3./ n nc4 nc6 n 3 C nc6 e i, then ja 0;3 j nc6 < ja n ;3j, we obtain By using this manner, we claim that p k./ p 4./ a n ;3q 3./ a 0;3 q3./ q 3./.n C 4/.n C 8/.n C 6/ Œn C.k /.n C k/ Œn C.k / nc6 e i q3./ n 3 n C 6 n C 8 n 4 C n C 8 e i n kc n C.k / n C k where k ; 3; ; n. To prove the equation (6) is correct for all k N C.k /, it suffices to show p kc./ ŒnC.k / ŒnC.kC/ n k n C k.n C k/ nc.kc/ n n k C n C k e i (6).kC/ C nc.kc/ e i (7) ownload ate //8 84 AM
7 Convolutions of harmonic right half-plane mappings 795 n kc nc.k / Let q k./ n k C nck nck e i, then q k./ n kc nc.k / q k.=/ C nck nck e i n kc. Since ja 0;k j j nck e i j nck < ja n kc;kj, by using Cohn s Rule on q k./, we deduce that p kc./ a n kc;kq k./ a 0;k q k./ q k./ ŒnC.k / ŒnC.kC/.n C k/ nck e i q k./ n k n C k nc.kc/ n.kc/ C nc.kc/ e i Setting n k.k / in (6), we have p n./ 3n.3n 4/.3n /! 3n e i 3n Then 0 3n e i 3n is a ero of p n./, and 3n e i j 0 j ˇ 3n ˇ j3n j C je i j 3n 3n C 3n So 0 lies inside or on the unit circle jj, by Lemma, we know that all eros of (5) lie on U. This completes the proof. Proof of Theorem.. In view of Lemma.6, it suffices to show that L c f a is locally univalent and sensepreserving. Substituting!./! a./.a /=. a/ into (0), we have e!./ Œ. c/. C c/ a a. C a a / c.a /. a/. / Œ. C c/. c/. C a a / c.a /. a/. / Œ. c/. C c/ Œ.a /. a/ C.a /.a /c. / Œ. C c/. c/ Œ. a/ C.a /. a/.a /c. / 3 Ca ccac Ca Cc C ccac Cc C Ca ccac a. c/ Cc Cc Ca ccac Cc a. c/ Cc 3 (8) Next we just need to show that je!./j < for 0 < c. C a/=. a/, where < a <. We shall consider the following two cases. Case. Suppose that a 0. Then substituting a 0 into (8) yields e!./ c Cc C c. / Cc c Cc C c Cc. / c Cc c Cc Then two eros and. c/=.cc/ of the above numerator lie in or on the unit circle for all 0 < c, so we have je!./j <. Case. Suppose that a 0. From (8), we can write e!./ 3 Ca ccac Ca Cc C ccac Cc C Ca ccac a. c/ Cc Cc Ca ccac Cc a. c/ Cc p./ 3 p./. A/. B/. C /. A/. B/. C / We will show that A; B; C U for 0 < c. C a/=. a/. Applying Lemma to p./ 3 C a c C ac C c C C a c C ac C c a. c/ C c ownload ate //8 84 AM
8 796 Y. Li, Z. Liu Note that j a. c/ j < for c > 0 and < a <, we get Cc p./ a 3p./ a 0 p a. c/./ p./ C Cc p./. C c C a ac/. C c a C ac/ C c 6ac C c Ca a c a c. C c/. C c/ C c C 6ac c a a c C a c. C c/. C c C a ac/. C c a C ac/ Cc a ac cca Cac C C. C c/ C c C a ac C c C a ac. C c C a ac/. C c a C ac/ C a c. a/. /. C c/ C a C c. a/ So p./ has two eros and Ca c. a/ which are in or on the unit circle for 0 < c. C CaCc. a/ a/=. a/. Thus, by Lemma, all eros of p./ lie on U, that is A; B; C U and so je!./j < for all U. 4 Examples In this section, we give interesting examples resulting from Theorem. and Theorem.. Example 4.. In Theorem., note that f h C g SH 0.a C /=. C a/. By shearing with h C g =. / and the dilatation!./ Solving these equations, we get h 0./ C g 0./. / and g 0./!./h 0./ h 0./ C a. / 3. C / and g 0./.a C /. / 3. C / Integration gives and then, By the convolutions, we have So that and h./ C C a 4 g./ C a 4. / C a log 8 a log. / 8 f 0 f h 0 h C g 0 g h./ C h0./ h 0 h 4 C C a 8 g 0 g 4 C a 8. / C a 6 log. / a 6 log g./ C C C C ; (9) C (0) g0./ C. C a/. / 3. C / C.a C /. / 3. C / Images of U under f 0 f for certain values of a are drawn in Figure and Figure (a)-(d) by using Mathematica. By Theorem., it follows that f 0 f is locally univalent and convex in the horiontal direction for a ; 05; 0;, but it is not locally univalent for a 0. ownload ate //8 84 AM
9 Convolutions of harmonic right half-plane mappings 797 Fig.. Images of U under f 0 f for a (a) a 0 (b) Partial enlarged view for a 0 Fig.. Images of U under f 0 f for various values a. (a) a (b) a 05 (c) a 0 (d) a ownload ate //8 84 AM
10 798 Y. Li, Z. Liu Example 4.. In Theorem., by () and (4), we have L c f a ha./ C ch 0 a C c./ C C c Re " Ca. / C. C c/. / C c ga./ c. a/. C a/. / 3 # c. a/. C / C. C c/. C a/. / 3 cg 0 a./ " C a Ca C c 8 #. /. C a/. / 3 c.a / < a Ca C c 9 = C i Im. C c/. / ; If we take a 05; c 6, in view of Theorem., we know that L 6 f 05 is univalent and convex in the horiontal direction. The image of U under f 05 ; L 6 and L 6 f 05 are shown in Figure 3 (a)-(c), respectively. Fig. 3. Images of U under f 05 ; L 6 and L 6 f (a) f (b) L (c) L 6 f 05 If we take a 05; c 6, then L 6 f 05 Re 7 C 3 08 (. C / 3 C i Im C 6. / 3 7. / The image of U under L 6 f 05 is shown in Figure 4 (a). Figure 4 (b) is a partial enlarged view of Figure 4 (a) showing that the images of two outer most concentric circles in U are intersecting and so L 6 f 05 is not univalent. ) ownload ate //8 84 AM
11 Convolutions of harmonic right half-plane mappings 799 Fig. 4. Images of the unit disk U under L 6 f (a) L 6 f 05 (b) Partial enlarged view of L 6 f 05 Remark 4.3. This example shows that the condition 0 < c. C a/=. a/ in Theorem. is sharp. Acknowledgement The authors would like to thank the referees for their helpful comments. According to the hints of the referees the authors were able to improve the paper considerably. The research was supported by the Project Education Fund of Yunnan Province under Grant No. 05Y456, the First Bath of Young and Middle-aged Academic Training Object Backbone of Honghe University under Grant No. 04GG00. References [] Kumar R., orff M., Gupta S., Singh S., Convolution properties of some harmonic mappings in the right-half plane, Bull. Malays. Math. Sci. Soc., 06, 39(), [] Kumar R., Gupta S., Singh S., orff M., On harmonic convolutions involving a vertical strip mapping, Bull. Korean Math. Soc., 05, 5(), 05 3 [3] Li L., Ponnusamy S., Convolutions of slanted half-plane harmonic mappings, Analysis (Munich), 03, 33(), [4] Li L., Ponnusamy S., Convolutions of harmonic mappings convex in one direction, Complex Anal. Oper. Theory, 05, 9(), [5] Boyd Z., orff M., Nowak M., Romney M., Wołoskiewic M., Univalency of convolutions of harmonic mappings, Appl. Math. Comput., 04, 34, [6] Liu Z., Li Y., The properties of a new subclass of harmonic univalent mappings, Abstr. Appl. Anal., 03, Article I 79408, 7 pages [7] Muir S., Weak subordination for convex univalent harmonic functions, J. Math. Anal. Appl., 008, 348, [8] Muir S., Convex combinations of planar harmonic mappings realied through convolutions with half-strip mappings, Bull. Malays. Math. Sci. Soc., 06, OI0.007/s [9] Nagpal S., Ravichandran V., Univalence and convexity in one direction of the convolution of harmonic mappings, Complex Var. Elliptic Equ., 04, 59(9), [0] Rosihan M. Ali., Stephen B. Adolf., Subramanian, K. G., Subclasses of harmonic mappings defined by convolution, Appl. Math. Lett., 00, 3, [] Wang Z., Liu Z., Li Y., On convolutions of harmonic univalent mappings convex in the direction of the real axis, J. Appl. Anal. Comput., 06, 6, [] Sokół Janus, Ibrahim Rabha W., Ahmad M. Z., Al-Janaby, Hiba F., Inequalities of harmonic univalent functions with connections of hypergeometric functions, Open Math., 05, 3, [3] Kumar R., Gupta S., Singh S., orff M., An application of Cohn s rule to convolutions of univalent harmonic mappings, arxiv v [4] orff M., Convolutions of planar harmonic convex mappins, Complex Var. Theorey Appl., 00, 45, 63 7 ownload ate //8 84 AM
12 800 Y. Li, Z. Liu [5] orff M., Nowak M., Wołoskiewic M., Convolutions of harmonic convex mappings, Complex Var. Elliptic Equ., 0, 57(5), [6] Li L., Ponnusamy S., Solution to an open problem on convolutions of harmonic mappings, Complex Var. Elliptic Equ., 03, 58(), [7] Li L., Ponnusamy S., Sections of stable harmonic convex functions, Nonlinear Anal., 05, 3-4, [8] uren P., Harmonic mappings in the Plane, Cambridge Tracts in Mathematics 56, Cambridge Univ. Press, Cambridge, 004 [9] Rahman Q. T., Schmeisser G., Analytic theory of polynomials, London Mathematical Society Monigraphs New Series 6, Oxford Univ. Press, Oxford, 00 [0] Clunie J., Sheil-Small T., Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A I. Math., 984, 9, 3 5 [] Pommenrenke C., On starlike and close-to-convex functions, Proc. London Math. Soc., 963, 3, ownload ate //8 84 AM
Convolution Properties of Convex Harmonic Functions
Int. J. Open Problems Complex Analysis, Vol. 4, No. 3, November 01 ISSN 074-87; Copyright c ICSRS Publication, 01 www.i-csrs.org Convolution Properties of Convex Harmonic Functions Raj Kumar, Sushma Gupta
More informationON POLYHARMONIC UNIVALENT MAPPINGS
ON POLYHARMONIC UNIVALENT MAPPINGS J. CHEN, A. RASILA and X. WANG In this paper, we introduce a class of complex-valued polyharmonic mappings, denoted by HS pλ, and its subclass HS 0 pλ, where λ [0, ]
More informationarxiv: v1 [math.cv] 28 Mar 2017
On third Hankel determinants for subclasses of analytic functions and close-to-convex harmonic mappings arxiv:703.09485v math.cv] 28 Mar 207 Yong Sun, Zhi-Gang Wang 2 and Antti Rasila 3 School of Science,
More informationINTEGRAL MEANS AND COEFFICIENT ESTIMATES ON PLANAR HARMONIC MAPPINGS
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 37 69 79 INTEGRAL MEANS AND COEFFICIENT ESTIMATES ON PLANAR HARMONIC MAPPINGS Shaolin Chen Saminathan Ponnusamy and Xiantao Wang Hunan Normal University
More informationON CERTAIN CLASSES OF UNIVALENT MEROMORPHIC FUNCTIONS ASSOCIATED WITH INTEGRAL OPERATORS
TWMS J. App. Eng. Math. V.4, No.1, 214, pp. 45-49. ON CERTAIN CLASSES OF UNIVALENT MEROMORPHIC FUNCTIONS ASSOCIATED WITH INTEGRAL OPERATORS F. GHANIM 1 Abstract. This paper illustrates how some inclusion
More informationRadius of close-to-convexity and fully starlikeness of harmonic mappings
Complex Variables and Elliptic Equations, 014 Vol. 59, No. 4, 539 55, http://dx.doi.org/10.1080/17476933.01.759565 Radius of close-to-convexity and fully starlikeness of harmonic mappings David Kalaj a,
More informationHARMONIC CLOSE-TO-CONVEX MAPPINGS
Applied Mathematics Stochastic Analysis, 15:1 (2002, 23-28. HARMONIC CLOSE-TO-CONVEX MAPPINGS JAY M. JAHANGIRI 1 Kent State University Department of Mathematics Burton, OH 44021-9500 USA E-mail: jay@geauga.kent.edu
More informationHarmonic Mappings and Shear Construction. References. Introduction - Definitions
Harmonic Mappings and Shear Construction KAUS 212 Stockholm, Sweden Tri Quach Department of Mathematics and Systems Analysis Aalto University School of Science Joint work with S. Ponnusamy and A. Rasila
More informationDIFFERENTIAL OPERATOR GENERALIZED BY FRACTIONAL DERIVATIVES
Miskolc Mathematical Notes HU e-issn 1787-2413 Vol. 12 (2011), No. 2, pp. 167 184 DIFFERENTIAL OPERATOR GENERALIZED BY FRACTIONAL DERIVATIVES RABHA W. IBRAHIM AND M. DARUS Received March 30, 2010 Abstract.
More informationdu+ z f 2(u) , which generalized the result corresponding to the class of analytic functions given by Nash.
Korean J. Math. 24 (2016), No. 3, pp. 36 374 http://dx.doi.org/10.11568/kjm.2016.24.3.36 SHARP HEREDITARY CONVEX RADIUS OF CONVEX HARMONIC MAPPINGS UNDER AN INTEGRAL OPERATOR Xingdi Chen and Jingjing Mu
More informationON CERTAIN SUBCLASSES OF UNIVALENT FUNCTIONS AND RADIUS PROPERTIES
ON CERTAIN SUBCLASSES OF UNIVALENT FUNCTIONS AND RADIUS PROPERTIES M. OBRADOVIĆ and S. PONNUSAMY Let S denote the class of normalied univalent functions f in the unit disk. One of the problems addressed
More informationJournal of Mathematical Analysis and Applications
J. Math. Anal. Appl. 373 2011 102 110 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa Bloch constant and Landau s theorem for planar
More informationarxiv: v3 [math.cv] 11 Aug 2014
File: SahSha-arXiv_Aug2014.tex, printed: 27-7-2018, 3.19 ON MAXIMAL AREA INTEGRAL PROBLEM FOR ANALYTIC FUNCTIONS IN THE STARLIKE FAMILY S. K. SAHOO AND N. L. SHARMA arxiv:1405.0469v3 [math.cv] 11 Aug 2014
More informationSUBCLASS OF HARMONIC STARLIKE FUNCTIONS ASSOCIATED WITH SALAGEAN DERIVATIVE
LE MATEMATICHE Vol. LXIX (2014) Fasc. II, pp. 147 158 doi: 10.4418/2014.69.2.13 SUBCLASS OF HARMONIC STARLIKE FUNCTIONS ASSOCIATED WITH SALAGEAN DERIVATIVE H. E. DARWISH - A. Y. LASHIN - S. M. SOILEH The
More informationWeak Subordination for Convex Univalent Harmonic Functions
Weak Subordination for Convex Univalent Harmonic Functions Stacey Muir Abstract For two complex-valued harmonic functions f and F defined in the open unit disk with f() = F () =, we say f is weakly subordinate
More informationSUBCLASSES OF P-VALENT STARLIKE FUNCTIONS DEFINED BY USING CERTAIN FRACTIONAL DERIVATIVE OPERATOR
Sutra: International Journal of Mathematical Science Education Technomathematics Research Foundation Vol. 4 No. 1, pp. 17-32, 2011 SUBCLASSES OF P-VALENT STARLIKE FUNCTIONS DEFINED BY USING CERTAIN FRACTIONAL
More informationConvolutions of Certain Analytic Functions
Proceedings of the ICM2010 Satellite Conference International Workshop on Harmonic and Quasiconformal Mappings (HQM2010) Editors: D. Minda, S. Ponnusamy, and N. Shanmugalingam J. Analysis Volume 18 (2010),
More informationOn Starlike and Convex Functions with Respect to 2k-Symmetric Conjugate Points
Tamsui Oxford Journal of Mathematical Sciences 24(3) (28) 277-287 Aletheia University On Starlike and Convex Functions with espect to 2k-Symmetric Conjugate Points Zhi-Gang Wang and Chun-Yi Gao College
More informationSome basic properties of certain subclasses of meromorphically starlike functions
Wang et al. Journal of Inequalities Applications 2014, 2014:29 R E S E A R C H Open Access Some basic properties of certain subclasses of meromorphically starlike functions Zhi-Gang Wang 1*, HM Srivastava
More informationCoefficient Estimates and Bloch s Constant in Some Classes of Harmonic Mappings
Bull. Malays. Math. Sci. Soc. (06) 39:74 750 DOI 0.007/s40840-05-038-9 Coefficient Estimates and Bloch s Constant in Some Classes of Harmonic Mappings S. Kanas D. Klimek-Smȩt Received: 5 September 03 /
More informationSubordination and Superordination Results for Analytic Functions Associated With Convolution Structure
Int. J. Open Problems Complex Analysis, Vol. 2, No. 2, July 2010 ISSN 2074-2827; Copyright c ICSRS Publication, 2010 www.i-csrs.org Subordination and Superordination Results for Analytic Functions Associated
More informationA Class of Univalent Harmonic Mappings
Mathematica Aeterna, Vol. 6, 016, no. 5, 675-680 A Class of Univalent Harmonic Mappings Jinjing Qiao Department of Mathematics, Hebei University, Baoding, Hebei 07100, People s Republic of China Qiannan
More informationA NEW CLASS OF MEROMORPHIC FUNCTIONS RELATED TO CHO-KWON-SRIVASTAVA OPERATOR. F. Ghanim and M. Darus. 1. Introduction
MATEMATIQKI VESNIK 66, 1 14, 9 18 March 14 originalni nauqni rad research paper A NEW CLASS OF MEROMORPHIC FUNCTIONS RELATED TO CHO-KWON-SRIVASTAVA OPERATOR F. Ghanim and M. Darus Abstract. In the present
More informationUDC S. Yu. Graf ON THE SCHWARZIAN NORM OF HARMONIC MAPPINGS
0 Probl. Anal. Issues Anal. Vol. 5 3), No., 016, pp. 0 3 DOI: 10.15393/j3.art.016.3511 UDC 517.54 S. Yu. Graf ON THE SCHWARZIAN NORM OF HARMONIC MAPPINGS Abstract. We obtain estimations of the pre-schwarzian
More informationFekete-Szegö Inequality for Certain Classes of Analytic Functions Associated with Srivastava-Attiya Integral Operator
Applied Mathematical Sciences, Vol. 9, 015, no. 68, 3357-3369 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ams.015.543 Fekete-Szegö Inequality for Certain Classes of Analytic Functions Associated
More informationA linear operator of a new class of multivalent harmonic functions
International Journal of Scientific Research Publications, Volume 7, Issue 8, August 2017 278 A linear operator of a new class of multivalent harmonic functions * WaggasGalibAtshan, ** Ali Hussein Battor,
More informationSUBCLASS OF HARMONIC UNIVALENT FUNCTIONS ASSOCIATED WITH THE DERIVATIVE OPERATOR
Hacettepe Journal of Mathematics and Statistics Volume 41(1) (2012), 47 58 SUBCLASS OF HARMONIC UNIVALENT FUNCTIONS ASSOCIATED WITH THE DERIVATIVE OPERATOR A.L. Pathak, S.B. Joshi, Preeti Dwivedi and R.
More informationThe Inner Mapping Radius of Harmonic Mappings of the Unit Disk 1
The Inner Mapping Radius of Harmonic Mappings of the Unit Disk Michael Dorff and Ted Suffridge Abstract The class S H consists of univalent, harmonic, and sense-preserving functions f in the unit disk,,
More informationResearch Article A New Class of Meromorphic Functions Associated with Spirallike Functions
Applied Mathematics Volume 202, Article ID 49497, 2 pages doi:0.55/202/49497 Research Article A New Class of Meromorphic Functions Associated with Spirallike Functions Lei Shi, Zhi-Gang Wang, and Jing-Ping
More informationAn Application of Wilf's Subordinating Factor Sequence on Certain Subclasses of Analytic Functions
International Journal of Applied Engineering Research ISS 0973-4562 Volume 3 umber 6 (208 pp. 2494-2500 An Application of Wilf's Subordinating Factor Sequence on Certain Subclasses of Analytic Functions
More informationarxiv: v1 [math.cv] 19 Jul 2012
arxiv:1207.4529v1 [math.cv] 19 Jul 2012 On the Radius Constants for Classes of Analytic Functions 1 ROSIHAN M. ALI, 2 NAVEEN KUMAR JAIN AND 3 V. RAVICHANDRAN 1,3 School of Mathematical Sciences, Universiti
More informationDIFFERENTIAL SUBORDINATION AND SUPERORDINATION OF ANALYTIC FUNCTIONS DEFINED BY THE MULTIPLIER TRANSFORMATION
M athematical I nequalities & A pplications Volume 12, Number 1 2009), 123 139 DIFFERENTIAL SUBORDINATION AND SUPERORDINATION OF ANALYTIC FUNCTIONS DEFINED BY THE MULTIPLIER TRANSFORMATION ROSIHAN M. ALI,
More informationResearch Article A Third-Order Differential Equation and Starlikeness of a Double Integral Operator
Abstract and Applied Analysis Volume 2, Article ID 9235, pages doi:.55/2/9235 Research Article A Third-Order Differential Equation and Starlikeness of a Double Integral Operator Rosihan M. Ali, See Keong
More informationAN EXTENSION OF THE REGION OF VARIABILITY OF A SUBCLASS OF UNIVALENT FUNCTIONS
AN EXTENSION OF THE REGION OF VARIABILITY OF A SUBCLASS OF UNIVALENT FUNCTIONS SUKHWINDER SINGH SUSHMA GUPTA AND SUKHJIT SINGH Department of Applied Sciences Department of Mathematics B.B.S.B. Engineering
More informationRosihan M. Ali and V. Ravichandran 1. INTRODUCTION
TAIWANESE JOURNAL OF MATHEMATICS Vol. 14, No. 4, pp. 1479-1490, August 2010 This paper is available online at http://www.tjm.nsysu.edu.tw/ CLASSES OF MEROMORPHIC α-convex FUNCTIONS Rosihan M. Ali and V.
More informationCesàro Sum Approximation of Outer Functions. R.W. Barnard, K. Pearce
Cesàro Sum Approximation of Outer Functions R.W. Barnard, K. Pearce Department of Mathematics and Statistics Texas Tech University barnard@math.ttu.edu, pearce@math.ttu.edu J. Cima Department of Mathematics
More informationStarlike Functions of Complex Order
Applied Mathematical Sciences, Vol. 3, 2009, no. 12, 557-564 Starlike Functions of Complex Order Aini Janteng School of Science and Technology Universiti Malaysia Sabah, Locked Bag No. 2073 88999 Kota
More informationDifferential Operator of a Class of Meromorphic Univalent Functions With Negative Coefficients
Differential Operator of a Class of Meromorphic Univalent Functions With Negative Coefficients Waggas Galib Atshan and Ali Hamza Abada Department of Mathematics College of Computer Science and Mathematics
More informationOn close-to-convex functions satisfying a differential inequality
Stud. Univ. Babeş-Bolyai Math. 60(2015), No. 4, 561 565 On close-to-convex functions satisfying a differential inequality Sukhwinder Singh Billing Abstract. Let C α(β) denote the class of normalized functions
More informationOn Quasi-Hadamard Product of Certain Classes of Analytic Functions
Bulletin of Mathematical analysis Applications ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 1, Issue 2, (2009), Pages 36-46 On Quasi-Hadamard Product of Certain Classes of Analytic Functions Wei-Ping
More informationDifferential subordination theorems for new classes of meromorphic multivalent Quasi-Convex functions and some applications
Int. J. Adv. Appl. Math. and Mech. 2(3) (2015) 126-133 (ISSN: 2347-2529) Journal homepage: www.ijaamm.com International Journal of Advances in Applied Mathematics and Mechanics Differential subordination
More informationON A SUBCLASS OF HARMONIC UNIVALENT FUNCTIONS DEFINED BY CONVOLUTION AND INTEGRAL CONVOLUTION
International Journal of Pure and pplied Mathematics Volume 69 No. 3 2011, 255-264 ON SUBCLSS OF HRMONIC UNIVLENT FUNCTIONS DEFINED BY CONVOLUTION ND INTEGRL CONVOLUTION K.K. Dixit 1,.L. Patha 2, S. Porwal
More informationSome applications of differential subordinations in the geometric function theory
Sokół Journal of Inequalities and Applications 2013, 2013:389 R E S E A R C H Open Access Some applications of differential subordinations in the geometric function theory Janus Sokół * Dedicated to Professor
More informationDifferential Subordinations and Harmonic Means
Bull. Malays. Math. Sci. Soc. (2015) 38:1243 1253 DOI 10.1007/s40840-014-0078-9 Differential Subordinations and Harmonic Means Stanisława Kanas Andreea-Elena Tudor eceived: 9 January 2014 / evised: 3 April
More informationTHE FEKETE-SZEGÖ COEFFICIENT FUNCTIONAL FOR TRANSFORMS OF ANALYTIC FUNCTIONS. Communicated by Mohammad Sal Moslehian. 1.
Bulletin of the Iranian Mathematical Society Vol. 35 No. (009 ), pp 119-14. THE FEKETE-SZEGÖ COEFFICIENT FUNCTIONAL FOR TRANSFORMS OF ANALYTIC FUNCTIONS R.M. ALI, S.K. LEE, V. RAVICHANDRAN AND S. SUPRAMANIAM
More informationThe Noor Integral and Strongly Starlike Functions
Journal of Mathematical Analysis and Applications 61, 441 447 (001) doi:10.1006/jmaa.001.7489, available online at http://www.idealibrary.com on The Noor Integral and Strongly Starlike Functions Jinlin
More informationHarmonic multivalent meromorphic functions. defined by an integral operator
Journal of Applied Mathematics & Bioinformatics, vol.2, no.3, 2012, 99-114 ISSN: 1792-6602 (print), 1792-6939 (online) Scienpress Ltd, 2012 Harmonic multivalent meromorphic functions defined by an integral
More informationOn Multivalent Functions Associated with Fixed Second Coefficient and the Principle of Subordination
International Journal of Mathematical Analysis Vol. 9, 015, no. 18, 883-895 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ijma.015.538 On Multivalent Functions Associated with Fixed Second Coefficient
More informationRosihan M. Ali, M. Hussain Khan, V. Ravichandran, and K. G. Subramanian. Let A(p, m) be the class of all p-valent analytic functions f(z) = z p +
Bull Korean Math Soc 43 2006, No 1, pp 179 188 A CLASS OF MULTIVALENT FUNCTIONS WITH NEGATIVE COEFFICIENTS DEFINED BY CONVOLUTION Rosihan M Ali, M Hussain Khan, V Ravichandran, and K G Subramanian Abstract
More informationBOUNDEDNESS, UNIVALENCE AND QUASICONFORMAL EXTENSION OF ROBERTSON FUNCTIONS. Ikkei Hotta and Li-Mei Wang
Indian J. Pure Appl. Math., 42(4): 239-248, August 2011 c Indian National Science Academy BOUNDEDNESS, UNIVALENCE AND QUASICONFORMAL EXTENSION OF ROBERTSON FUNCTIONS Ikkei Hotta and Li-Mei Wang Department
More informationSecond Hankel determinant for the class of Bazilevic functions
Stud. Univ. Babeş-Bolyai Math. 60(2015), No. 3, 413 420 Second Hankel determinant for the class of Bazilevic functions D. Vamshee Krishna and T. RamReddy Abstract. The objective of this paper is to obtain
More informationDIFFERENTIAL SUBORDINATION ASSOCIATED WITH NEW GENERALIZED DERIVATIVE OPERATOR
ROMAI J., v.12, no.1(2016), 77 89 DIFFERENTIAL SUBORDINATION ASSOCIATED WITH NEW GENERALIZED DERIVATIVE OPERATOR Anessa Oshah, Maslina Darus School of Mathematical Sciences, Faculty of Science and Technology,
More informationOn the improvement of Mocanu s conditions
Nunokawa et al. Journal of Inequalities Applications 13, 13:46 http://www.journalofinequalitiesapplications.com/content/13/1/46 R E S E A R C H Open Access On the improvement of Mocanu s conditions M Nunokawa
More informationn=2 AMS Subject Classification: 30C45. Keywords and phrases: Starlike functions, close-to-convex functions, differential subordination.
MATEMATIQKI VESNIK 58 (2006), 119 124 UDK 517.547 originalni nauqni rad research paper ON CETAIN NEW SUBCLASS OF CLOSE-TO-CONVEX FUNCTIONS Zhi-Gang Wang, Chun-Yi Gao and Shao-Mou Yuan Abstract. In the
More informationdenote the subclass of the functions f H of the form H of the form
Scholars Journal of Engineering and Technology (SJET) Sch. J. Eng. Tech., 05; 3(4C):55-59 Scholars Academic and Scientific Publisher (An International Publisher for Academic and Scientific Resources) www.saspublisher.com
More informationResearch Article On an Integral Transform of a Class of Analytic Functions
Abstract and Applied Analysis Volume 22, Article ID 25954, pages doi:.55/22/25954 Research Article On an Integral Transform of a Class of Analytic Functions Sarika Verma, Sushma Gupta, and Sukhjit Singh
More informationResearch Article Coefficient Conditions for Harmonic Close-to-Convex Functions
Abstract and Applied Analysis Volume 212, Article ID 413965, 12 pages doi:1.1155/212/413965 Research Article Coefficient Conditions for Harmonic Close-to-Convex Functions Toshio Hayami Department of Mathematics,
More informationON CERTAIN CLASS OF UNIVALENT FUNCTIONS WITH CONIC DOMAINS INVOLVING SOKÓ L - NUNOKAWA CLASS
U.P.B. Sci. Bull., Series A, Vol. 80, Iss. 1, 018 ISSN 13-707 ON CERTAIN CLASS OF UNIVALENT FUNCTIONS WITH CONIC DOMAINS INVOLVING SOKÓ L - NUNOKAWA CLASS S. Sivasubramanian 1, M. Govindaraj, K. Piejko
More informationA NEW SUBCLASS OF MEROMORPHIC FUNCTION WITH POSITIVE COEFFICIENTS
Bulletin of Mathematical Analysis and Applications ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 3 Issue 32010), Pages 109-121. A NEW SUBCLASS OF MEROMORPHIC FUNCTION WITH POSITIVE COEFFICIENTS S.
More informationLinear functionals and the duality principle for harmonic functions
Math. Nachr. 285, No. 13, 1565 1571 (2012) / DOI 10.1002/mana.201100259 Linear functionals and the duality principle for harmonic functions Rosihan M. Ali 1 and S. Ponnusamy 2 1 School of Mathematical
More informationOn Classes of Analytic Functions Associated by a Parametric Linear Operator in a Complex Domain
Proceedings of the Pakistan Academy of Sciences: A. Physical and Computational Sciences 54 (3): 257 267 (207) Copyright Pakistan Academy of Sciences ISSN: 258-4245 (print), 258-4253 (online) Pakistan Academy
More informationa n z n, z U.. (1) f(z) = z + n=2 n=2 a nz n and g(z) = z + (a 1n...a mn )z n,, z U. n=2 a(a + 1)b(b + 1) z 2 + c(c + 1) 2! +...
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.13(2012) No.2,pp.153-157 Integral Operator Defined by Convolution Product of Hypergeometric Functions Maslina Darus,
More informationInitial Coefficient Bounds for a General Class of Bi-Univalent Functions
Filomat 29:6 (2015), 1259 1267 DOI 10.2298/FIL1506259O Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://.pmf.ni.ac.rs/filomat Initial Coefficient Bounds for
More informationA Certain Subclass of Analytic Functions Defined by Means of Differential Subordination
Filomat 3:4 6), 3743 3757 DOI.98/FIL64743S Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat A Certain Subclass of Analytic Functions
More informationNOTE ON RADIUS PROBLEMS FOR CERTAIN CLASS OF ANALYTIC FUNCTIONS. We dedicate this paper to the 60th anniversary of Professor Y. Polatoglu.
NOTE ON RADIUS PROBLEMS FOR CERTAIN CLASS OF ANALYTIC FUNCTIONS Neslihan Uyanik 1, Shigeyoshi Owa 2 We dedicate this paper to the 60th anniversary of Professor Y. Polatoglu Abstract For analytic functions
More informationFABER POLYNOMIAL COEFFICIENT ESTIMATES FOR A NEW SUBCLASS OF MEROMORPHIC BI-UNIVALENT FUNCTIONS ADNAN GHAZY ALAMOUSH, MASLINA DARUS
Available online at http://scik.org Adv. Inequal. Appl. 216, 216:3 ISSN: 25-7461 FABER POLYNOMIAL COEFFICIENT ESTIMATES FOR A NEW SUBCLASS OF MEROMORPHIC BI-UNIVALENT FUNCTIONS ADNAN GHAZY ALAMOUSH, MASLINA
More informationCoefficient bounds for some subclasses of p-valently starlike functions
doi: 0.2478/v0062-02-0032-y ANNALES UNIVERSITATIS MARIAE CURIE-SKŁODOWSKA LUBLIN POLONIA VOL. LXVII, NO. 2, 203 SECTIO A 65 78 C. SELVARAJ, O. S. BABU G. MURUGUSUNDARAMOORTHY Coefficient bounds for some
More informationSOME INCLUSION PROPERTIES OF STARLIKE AND CONVEX FUNCTIONS ASSOCIATED WITH HOHLOV OPERATOR. II
italian journal of pure and applied mathematics n. 37 2017 117 126 117 SOME INCLUSION PROPERTIES OF STARLIKE AND CONVEX FUNCTIONS ASSOCIATED WITH HOHLOV OPERATOR. II M. Kasthuri K. Vijaya 1 K. Uma School
More informationDIFFERENTIAL SANDWICH THEOREMS FOR SOME SUBCLASSES OF ANALYTIC FUNCTIONS INVOLVING A LINEAR OPERATOR. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXXVI, 2(2007), pp. 287 294 287 DIFFERENTIAL SANDWICH THEOREMS FOR SOME SUBCLASSES OF ANALYTIC FUNCTIONS INVOLVING A LINEAR OPERATOR T. N. SHAMMUGAM, C. RAMACHANDRAN, M.
More informationSOME PROPERTIES OF A SUBCLASS OF ANALYTIC FUNCTIONS DEFINED BY A GENERALIZED SRIVASTAVA-ATTIYA OPERATOR. Nagat. M. Mustafa and Maslina Darus
FACTA UNIVERSITATIS NIŠ) Ser. Math. Inform. Vol. 27 No 3 2012), 309 320 SOME PROPERTIES OF A SUBCLASS OF ANALYTIC FUNCTIONS DEFINED BY A GENERALIZED SRIVASTAVA-ATTIYA OPERATOR Nagat. M. Mustafa and Maslina
More informationCERTAIN DIFFERENTIAL SUBORDINATIONS USING A GENERALIZED SĂLĂGEAN OPERATOR AND RUSCHEWEYH OPERATOR. Alina Alb Lupaş
CERTAIN DIFFERENTIAL SUBORDINATIONS USING A GENERALIZED SĂLĂGEAN OPERATOR AND RUSCHEWEYH OPERATOR Alina Alb Lupaş Dedicated to Prof. H.M. Srivastava for the 70th anniversary Abstract In the present paper
More informationOn a conjecture of S.P. Robinson
J. Math. Anal. Appl. 31 (005) 548 554 www.elsevier.com/locate/jmaa On a conjecture of S.P. Robinson Stephan Ruscheweyh a, Luis Salinas b, a Mathematisches Institut, Universität Würzburg, D-97074 Würzburg,
More informationInclusion relationships for certain classes of analytic functions involving the Choi-Saigo-Srivastava operator
Cho Yoon Journal of Inequalities Applications 2013, 2013:83 R E S E A R C H Open Access Inclusion relationships for certain classes of analytic functions involving the Choi-Saigo-Srivastava operator Nak
More informationOn a New Subclass of Meromorphically Multivalent Functions Defined by Linear Operator
International Mathematical Forum, Vol. 8, 213, no. 15, 713-726 HIKARI Ltd, www.m-hikari.com On a New Subclass of Meromorphically Multivalent Functions Defined by Linear Operator Ali Hussein Battor, Waggas
More informationNEW SUBCLASS OF MULTIVALENT HYPERGEOMETRIC MEROMORPHIC FUNCTIONS
Kragujevac Journal of Mathematics Volume 42(1) (2018), Pages 83 95. NEW SUBCLASS OF MULTIVALENT HYPERGEOMETRIC MEROMORPHIC FUNCTIONS M. ALBEHBAH 1 AND M. DARUS 2 Abstract. In this aer, we introduce a new
More informationFixed points of the derivative and k-th power of solutions of complex linear differential equations in the unit disc
Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 48, -9; http://www.math.u-szeged.hu/ejqtde/ Fixed points of the derivative and -th power of solutions of complex linear differential
More informationDIFFERENTIAL SANDWICH THEOREMS FOR SOME SUBCLASSES OF ANALYTIC FUNCTIONS INVOLVING A LINEAR OPERATOR. 1. Introduction
DIFFERENTIAL SANDWICH THEOREMS FOR SOME SUBCLASSES OF ANALYTIC FUNCTIONS INVOLVING A LINEAR OPERATOR T. N. SHAMMUGAM, C. RAMACHANDRAN, M. DARUS S. SIVASUBRAMANIAN Abstract. By making use of the familiar
More informationHAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM
Georgian Mathematical Journal Volume 9 (2002), Number 3, 591 600 NONEXPANSIVE MAPPINGS AND ITERATIVE METHODS IN UNIFORMLY CONVEX BANACH SPACES HAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM
More informationStrong convergence of multi-step iterates with errors for generalized asymptotically quasi-nonexpansive mappings
Palestine Journal of Mathematics Vol. 1 01, 50 64 Palestine Polytechnic University-PPU 01 Strong convergence of multi-step iterates with errors for generalized asymptotically quasi-nonexpansive mappings
More informationMapping problems and harmonic univalent mappings
Mapping problems and harmonic univalent mappings Antti Rasila Helsinki University of Technology antti.rasila@tkk.fi (Mainly based on P. Duren s book Harmonic mappings in the plane) Helsinki Analysis Seminar,
More informationSOME APPLICATIONS OF SALAGEAN INTEGRAL OPERATOR. Let A denote the class of functions of the form: f(z) = z + a k z k (1.1)
STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume LV, Number 1, March 2010 SOME APPLICATIONS OF SALAGEAN INTEGRAL OPERATOR Abstract. In this paper we introduce and study some new subclasses of starlike, convex,
More informationResearch Article A New Roper-Suffridge Extension Operator on a Reinhardt Domain
Abstract and Applied Analysis Volume 2011, Article ID 865496, 14 pages doi:10.1155/2011/865496 Research Article A New Roper-Suffridge Extension Operator on a Reinhardt Domain Jianfei Wang and Cailing Gao
More informationOn Toader-Sándor Mean 1
International Mathematical Forum, Vol. 8, 213, no. 22, 157-167 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/imf.213.3344 On Toader-Sándor Mean 1 Yingqing Song College of Mathematics and Computation
More informationCoefficient Bounds for a Certain Class of Analytic and Bi-Univalent Functions
Filomat 9:8 (015), 1839 1845 DOI 10.98/FIL1508839S Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Coefficient Bounds for a Certain
More informationSome Geometric Properties of a Certain Subclass of Univalent Functions Defined by Differential Subordination Property
Gen. Math. Notes Vol. 20 No. 2 February 2014 pp. 79-94 SSN 2219-7184; Copyright CSRS Publication 2014 www.i-csrs.org Available free online at http://www.geman.in Some Geometric Properties of a Certain
More informationCOEFFICIENT INEQUALITY FOR CERTAIN SUBCLASS OF ANALYTIC FUNCTIONS
Armenian Journal of Mathematics Volume 4, Number 2, 2012, 98 105 COEFFICIENT INEQUALITY FOR CERTAIN SUBCLASS OF ANALYTIC FUNCTIONS D Vamshee Krishna* and T Ramreddy** * Department of Mathematics, School
More informationFABER POLYNOMIAL COEFFICIENTS FOR GENERALIZED BI SUBORDINATE FUNCTIONS OF COMPLEX ORDER
Journal of Mathematical Inequalities Volume, Number 3 (08), 645 653 doi:0.753/jmi-08--49 FABER POLYNOMIAL COEFFICIENTS FOR GENERALIZED BI SUBORDINATE FUNCTIONS OF COMPLEX ORDER ERHAN DENIZ, JAY M. JAHANGIRI,
More informationThe Generalized Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces
Applied Mathematical Sciences, Vol. 11, 2017, no. 12, 549-560 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.718 The Generalized Viscosity Implicit Rules of Asymptotically Nonexpansive
More informationA PLANAR INTEGRAL SELF-AFFINE TILE WITH CANTOR SET INTERSECTIONS WITH ITS NEIGHBORS
#A INTEGERS 9 (9), 7-37 A PLANAR INTEGRAL SELF-AFFINE TILE WITH CANTOR SET INTERSECTIONS WITH ITS NEIGHBORS Shu-Juan Duan School of Math. and Computational Sciences, Xiangtan University, Hunan, China jjmmcn@6.com
More informationMalaya J. Mat. 4(1)(2016) 37-41
Malaya J. Mat. 4(1)(2016) 37-41 Certain coefficient inequalities for -valent functions Rahim Kargar a,, Ali Ebadian a and Janus Sokół b a Deartment of Mathematics, Payame Noor University, I. R. of Iran.
More informationA Formula for the Specialization of Skew Schur Functions
A Formula for the Specialization of Skew Schur Functions The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation As Published Publisher
More informationMajorization for Certain Classes of Meromorphic Functions Defined by Integral Operator
Int. J. Open Problems Complex Analysis, Vol. 5, No. 3, November, 2013 ISSN 2074-2827; Copyright c ICSRS Publication, 2013 www.i-csrs.org Majorization for Certain Classes of Meromorphic Functions Defined
More informationA general coefficient inequality 1
General Mathematics Vol. 17, No. 3 (2009), 99 104 A general coefficient inequality 1 Sukhwinder Singh, Sushma Gupta and Sukhjit Singh Abstract In the present note, we obtain a general coefficient inequality
More informationSOME CRITERIA FOR STRONGLY STARLIKE MULTIVALENT FUNCTIONS
Bulletin of the Institute of Mathematics Academia Sinica (New Series) Vol. 7 (01), No. 3, pp. 43-436 SOME CRITERIA FOR STRONGLY STARLIKE MULTIVALENT FUNCTIONS DINGGONG YANG 1 AND NENG XU,b 1 Department
More informationarxiv: v2 [math.cv] 24 Nov 2017
adii of the β uniformly convex of order α of Lommel and Struve functions Sercan Topkaya, Erhan Deniz and Murat Çağlar arxiv:70.0975v [math.cv] 4 Nov 07 Department of Mathematics, Faculty of Science and
More informationCoecient bounds for certain subclasses of analytic functions of complex order
Hacettepe Journal of Mathematics and Statistics Volume 45 (4) (2016), 1015 1022 Coecient bounds for certain subclasses of analytic functions of complex order Serap Bulut Abstract In this paper, we introduce
More information引用北海学園大学学園論集 (171): 11-24
タイトル 著者 On Some Singular Integral Operato One to One Mappings on the Weight Hilbert Spaces YAMAMOTO, Takanori 引用北海学園大学学園論集 (171): 11-24 発行日 2017-03-25 On Some Singular Integral Operators Which are One
More informationShih-sen Chang, Yeol Je Cho, and Haiyun Zhou
J. Korean Math. Soc. 38 (2001), No. 6, pp. 1245 1260 DEMI-CLOSED PRINCIPLE AND WEAK CONVERGENCE PROBLEMS FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS Shih-sen Chang, Yeol Je Cho, and Haiyun Zhou Abstract.
More informationImproved Newton s method with exact line searches to solve quadratic matrix equation
Journal of Computational and Applied Mathematics 222 (2008) 645 654 wwwelseviercom/locate/cam Improved Newton s method with exact line searches to solve quadratic matrix equation Jian-hui Long, Xi-yan
More informationResearch Article Subordination Results on Subclasses Concerning Sakaguchi Functions
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 574014, 7 pages doi:10.1155/2009/574014 Research Article Subordination Results on Subclasses Concerning Sakaguchi
More information