Abstract
By applying a fractional q-calculus operator, we define the subclasses S˛
n .;ˇ;b;q/
and G ˛
n .;ˇ;b;q/ of normalized analytic functions with complex order and negative coefficients.
Among the results investigated for each of these function classes, we derive their associated
coefficient estimates, radii of close-to-convexity, starlikeness and convexity, extreme points, and
growth and distortion theorems.
2010 Mathematics Subject Classification: 26A33; 30C45; 33D05
Results & Lemmas (30)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Theorem 1.
Theorem 1. The function f.´/ 2 S˛ n.;ˇ;b;q/ if and only if 1 X kDnC1 Œkq Cˇ jbj 1 Œ1C˛.Œkq 1/Aq.;k/ak 5 ˇ jbj: (2.2)
Theorem 1. The function f .´/ 2 S˛ n .;ˇ;b;q/ if and only if 1 X kDnC1 Œkq Cˇ jbj 1 Œ1C˛.Œkq 1/Aq.;k/ak 5 ˇ jbj: (2.2)
Corollary 1.
Corollary 1. Let f.´/ 2 S˛ n.;ˇ;b;q/. Then ak 5 ˇ jbj.Œkq Cˇ jbj 1/Œ1C˛.Œkq 1/Aq.;k/.k = nC1/: (2.4) The result is sharp for the…
Corollary 1. Let f .´/ 2 S˛ n .;ˇ;b;q/. Then ak 5 ˇ jbj .Œkq Cˇ jbj 1/Œ1C˛.Œkq 1/Aq.;k/ .k = nC1/: (2.4) The result is sharp for the function f .´/ given (for .k = nC1) by f .´/ D ´ ˇ jbj .Œkq Cˇ jbj 1/Œ1C˛.Œkq 1/Aq.;k/´k: (2.5) Putting ˇ D 1 in Theorem 1, we have Corollary 2 below.
Corollary 2.
Corollary 2. Let f.´/ 2 S˛ n.;b;q/. Then 1 X kDnC1.Œkq Cjbj 1/Œ1C˛.Œkq 1/Aq.;k/ak 5 jbj:
Corollary 2. Let f .´/ 2 S˛ n .;b;q/. Then 1 X kDnC1 .Œkq Cjbj 1/Œ1C˛.Œkq 1/Aq.;k/ak 5 jbj:
Corollary 3.
Corollary 3. Let f.´/ 2 S˛ n.;b;q/. Then ak 5 jbj.Œkq Cjbj 1/Œ1C˛.Œkq 1/Aq.;k/.k = nC1/: The result is sharp for the function f.´/…
Corollary 3. Let f .´/ 2 S˛ n .;b;q/. Then ak 5 jbj .Œkq Cjbj 1/Œ1C˛.Œkq 1/Aq.;k/ .k = nC1/: The result is sharp for the function f .´/ given by f .´/ D ´ jbj .Œkq Cjbj 1/Œ1C˛.Œkq 1/Aq.;k/ ´k .k = nC1/: It is not difficult to prove the following results. The details involved are being left as an exercise for the interested reader.
Theorem 2.
Theorem 2. The function f.´/ 2 G ˛ n.;ˇ;b;q/ if and only if 1 X kDnC1 Œkq 1C˛.Œkq 1/ Aq.;k/ak 5 ˇ jbj: (2.6)
Theorem 2. The function f .´/ 2 G ˛ n .;ˇ;b;q/ if and only if 1 X kDnC1 Œkq 1C˛.Œkq 1/ Aq.;k/ak 5 ˇ jbj: (2.6)
Corollary 4.
Corollary 4. Let f.´/ 2 G ˛ n.;ˇ;b;q/. Then ak 5 ˇ jbj Œkq 1C˛.Œkq 1/ Aq.;k/: (2.7) The result is sharp for the function f.´/…
Corollary 4. Let f .´/ 2 G ˛ n .;ˇ;b;q/. Then ak 5 ˇ jbj Œkq 1C˛.Œkq 1/ Aq.;k/: (2.7) The result is sharp for the function f .´/ given by f .´/ D ´ ˇ jbj Œkq 1C˛Œkq 1
Theorem 3.
Theorem 3. If b1;b2 2 C and jb1j < jb2j; then S˛ n.;ˇ;b1;q/ S˛ n.;ˇ;b2;q/: The following result can indeed be proven along the lines…
Theorem 3. If b1;b2 2 C and jb1j < jb2j; then S˛ n .;ˇ;b1;q/ S˛ n .;ˇ;b2;q/: The following result can indeed be proven along the lines which we have already indicated above.
Theorem 4.
Theorem 4. If b1;b2 2 C and jb1j < jb2j; then G ˛ n.;ˇ;b1;q/ G ˛ n.;ˇ;b2;q/: (2.9) 3. EXTREME POINTS FOR THE FUNCTION CLASSES S˛…
Theorem 4. If b1;b2 2 C and jb1j < jb2j; then G ˛ n .;ˇ;b1;q/ G ˛ n .;ˇ;b2;q/: (2.9) 3. EXTREME POINTS FOR THE FUNCTION CLASSES S˛ n .;ˇ;b;q/ AND G ˛ n .;ˇ;b;q/ In this section, we first prove the following result.
Theorem 5.
Theorem 5. Let fn.´/ D ´ and fk.´/ D ´ ˇ jbj.Œkq Cˇ jbj 1/Œ1C˛.Œkq 1/Aq.;k/´k (3.1).k = nC1/: Then the function f.´/ is in the class…
Theorem 5. Let fn.´/ D ´ and fk.´/ D ´ ˇ jbj .Œkq Cˇ jbj 1/Œ1C˛.Œkq 1/Aq.;k/´k (3.1) .k = nC1/: Then the function f .´/ is in the class S˛ n .;ˇ;b;q/ if and only if it can be expressed in the following formW f .´/ D 1 X kDn kfk.´/; (3.2) where
Corollary 5.
Corollary 5. The extreme points of the function class S˛ n.;ˇ;b;q/ are the func- tions fn.´/ D ´ and fk.´/.k = nC1/ given by (3.1).…
Corollary 5. The extreme points of the function class S˛ n .;ˇ;b;q/ are the func- tions fn.´/ D ´ and fk.´/ .k = nC1/ given by (3.1). Similarly, we can prove the following theorem.
Theorem 6.
Theorem 6. Let fn.´/ D ´ and fk.´/ D ´ ˇ jbj ŒkqŒ1C˛.Œkq 1/Aq.;k/´k.k = nC1/: (3.3)
Theorem 6. Let fn.´/ D ´ and fk.´/ D ´ ˇ jbj ŒkqŒ1C˛.Œkq 1/Aq.;k/´k .k = nC1/: (3.3)
Corollary 6.
Corollary 6. The extreme points of the function class G ˛ n.;ˇ;b;q/ are the func- tions fn.´/ D ´ and fk.´/.k = nC1/ given by (3.3). 4.…
Corollary 6. The extreme points of the function class G ˛ n .;ˇ;b;q/ are the func- tions fn.´/ D ´ and fk.´/ .k = nC1/ given by (3.3). 4. RADII OF CLOSE-TO-CONVEXITY, STARLIKENESS AND CONVEXITY OF THE FUNCTION CLASS S˛ n .;ˇ;b;q/
Theorem 7.
Theorem 7. Let f.´/ 2 S˛ n.;ˇ;b;q/: Then f.´/ is close-to-convex of order .0 5 < 1/ in j´j < r1; where r1 WD inf k=nC1 .1 /.Œkq Cˇ…
Theorem 7. Let f .´/ 2 S˛ n .;ˇ;b;q/: Then f .´/ is close-to-convex of order .0 5 < 1/ in j´j < r1; where r1 WD inf k=nC1 .1 /.Œkq Cˇ jbj 1/Œ1C˛.Œkq 1/Aq.;k/ kˇ jbj 1 k 1 : (4.1) The sharpness of this result is attained for the function f .´/ given by (2.5).
Theorem 8.
Theorem 8. Let f.´/ 2 S˛ n.;ˇ;b;q/: Then the function f.´/ is starlike of order .0 5 < 1/ in j´j < r2; where r2 WD inf k=nC1…
Theorem 8. Let f .´/ 2 S˛ n .;ˇ;b;q/: Then the function f .´/ is starlike of order .0 5 < 1/ in j´j < r2; where r2 WD inf k=nC1 .1 /.Œkq Cˇ jbj 1/Œ1C˛.Œkq 1/Aq.;k/ .k /ˇ jbj 1 k 1 : (4.3) The sharpness of this result is attained for the function f .´/ given by (2.5).
Corollary 7.
Corollary 7. Let f.´/ 2 S˛ n.;ˇ;b;q/: Then the function f.´/ is convex of order .0 5 < 1/ in j´j < r3; where r3 WD inf k=nC1…
Corollary 7. Let f .´/ 2 S˛ n .;ˇ;b;q/: Then the function f .´/ is convex of order .0 5 < 1/ in j´j < r3; where r3 WD inf k=nC1 .1 /.Œkq Cˇ jbj 1/Œ1C˛.Œkq 1/Aq.;k/ k.k /ˇ jbj 1 k 1 : The sharpness of the result is attained for the function f .´/ given by (2.5). 5. GROWTH AND DISTORTION THEOREMS For convenience in this section, for k = nC1; we shall henceforth use the follow-
Lemma 1.
Lemma 1. The sequence fAq.;k/g1 kDnC1 is a decreasing sequence in k.k = nC1/ for < 2 and 0 < q < 1:
Lemma 1. The sequence fAq.;k/g1 kDnC1 is a decreasing sequence in k .k = nC1/ for < 2 and 0 < q < 1:
Theorem 9.
Theorem 9. Let f.´/ 2 S˛ n.;ˇ;b;q/: Then j´j ˇ jbj nC1;˛.;ˇ;b;q/ j´jnC1 5 jf.´/j 5 j´jC ˇ jbj nC1;˛.;ˇ;b;q/ j´jnC1: (5.4) The result…
Theorem 9. Let f .´/ 2 S˛ n .;ˇ;b;q/: Then j´j ˇ jbj nC1;˛.;ˇ;b;q/ j´jnC1 5 jf .´/j 5 j´jC ˇ jbj nC1;˛.;ˇ;b;q/ j´jnC1 : (5.4) The result is sharp for the function f .´/ given by f .´/ D ´ ˇ jbj nC1;˛.;ˇ;b;q/´nC1: (5.5)
Corollary 8.
Corollary 8. Under the hypothesis of Theorem 9; the function f.´/ is included in a disk with center at the origin and radius r given by r D…
Corollary 8. Under the hypothesis of Theorem 9; the function f .´/ is included in a disk with center at the origin and radius r given by r D 1C ˇ jbj nC1;˛.;ˇ;b;q/: Similarly, we can prove the following distortion theorem for f .´/ 2 G ˛ n .;ˇ;b;q/:
Theorem 10.
Theorem 10. Let f.´/ 2 G ˛ n.;ˇ;b;q/ and let k;˛.;ˇ;b;q/ be given by (5.2): Then j´j ˇ jbj nC1;˛.;ˇ;b;q/ j´jnC1 5 jf.´/j 5 j´jC ˇ jbj…
Theorem 10. Let f .´/ 2 G ˛ n .;ˇ;b;q/ and let k;˛.;ˇ;b;q/ be given by (5.2): Then j´j ˇ jbj nC1;˛.;ˇ;b;q/ j´jnC1 5 jf .´/j 5 j´jC ˇ jbj nC1;˛.;ˇ;b;q/ j´jnC1 : (5.9) The result is sharp for the function f .´/ given by f .´/ D ´ ˇ jbj nC1;˛.;ˇ;b;q/´nC1: (5.10)
Corollary 9.
Corollary 9. Under the hypothesis of Theorem 10; the function f.´/ is included in a disk with its center at the origin and its radius r…
Corollary 9. Under the hypothesis of Theorem 10; the function f .´/ is included in a disk with its center at the origin and its radius r given by r D 1C ˇ jbj nC1;˛.;ˇ;b;q/: A further distortion theorem involving the generalized fractional q-differintegral operator ˝ q;´ defined by (1.7) is given by the following theorem.
Theorem 11.
Theorem 11. Let f.´/ 2 S˛ n.;ˇ;b;q/: Then j´j ˇ jbj.ŒnC1q Cˇ jbj 1/Œ1C˛.ŒnC1q 1/ j´jnC1 5 ˇˇˇ˝ q;´f.´/ ˇˇˇ 5 j´jC ˇ jbj.ŒnC1q Cˇ…
Theorem 11. Let f .´/ 2 S˛ n .;ˇ;b;q/: Then j´j ˇ jbj .ŒnC1q Cˇ jbj 1/Œ1C˛.ŒnC1q 1/ j´jnC1 5 ˇˇˇ˝ q;´f .´/ ˇˇˇ 5 j´jC ˇ jbj .ŒnC1q Cˇ jbj 1/Œ1C˛.ŒnC1q 1/ j´jnC1 : (5.11) The result is sharp.
Corollary 10.
Corollary 10. Let f.´/ 2 S˛ n.;ˇ;b;q/: Then q.2/ q.2 / j´j1 1 ˇ jbj.ŒnC1q Cˇ jbj 1/Œ1C˛.ŒnC1q 1/ j´jn 5 ˇˇˇD q;´f.´/ ˇˇˇ 5…
Corollary 10. Let f .´/ 2 S˛ n .;ˇ;b;q/: Then q.2/ q.2 / j´j1 1 ˇ jbj .ŒnC1q Cˇ jbj 1/Œ1C˛.ŒnC1q 1/ j´jn 5 ˇˇˇD q;´f .´/ ˇˇˇ 5 q.2/ q.2 / j´j1
Corollary 11.
Corollary 11. Let f.´/ 2 S˛ n.;b;q/: Then q.2/ q.2 / j´j1 1 jbj.ŒnC1q Cjbj 1/Œ1C˛.ŒnC1q 1/ j´jn 5 ˇˇˇD q;´f.´/ ˇˇˇ 5 q.2/…
Corollary 11. Let f .´/ 2 S˛ n .;b;q/: Then q.2/ q.2 / j´j1 1 jbj .ŒnC1q Cjbj 1/Œ1C˛.ŒnC1q 1/ j´jn 5 ˇˇˇD q;´f .´/ ˇˇˇ 5 q.2/ q.2 / j´j1
Corollary 12.
Corollary 12. Let f.´/ 2 S˛ n.;ˇ;b;q/: Then q.2/ q.2C/ j´j1C 1 ˇ jbj.ŒnC1q Cˇ jbj 1/Œ1C˛.ŒnC1q 1/ j´jn
Corollary 12. Let f .´/ 2 S˛ n .;ˇ;b;q/: Then q.2/ q.2C/ j´j1C 1 ˇ jbj .ŒnC1q Cˇ jbj 1/Œ1C˛.ŒnC1q 1/ j´jn
Corollary 13.
Corollary 13. Let f.´/ 2 S˛ n.;b;q/: Then q.2/ q.2C/ j´j1C 1 jbj.ŒnC1q Cjbj 1/Œ1C˛.ŒnC1q 1/ j´jn 5 ˇˇˇI q;´f.´/ ˇˇˇ 5 q.2/…
Corollary 13. Let f .´/ 2 S˛ n .;b;q/: Then q.2/ q.2C/ j´j1C 1 jbj .ŒnC1q Cjbj 1/Œ1C˛.ŒnC1q 1/ j´jn 5 ˇˇˇI q;´f .´/ ˇˇˇ 5 q.2/ q.2C/ j´j1C
Theorem 12.
Theorem 12. Let f.´/ 2 G ˛ n.;ˇ;b;q/: Then j´j ˇ jbj ŒnC1qŒ1C˛.ŒnC1q 1/ j´jnC1 5 ˇˇˇ˝ q;´f.´/ ˇˇˇ 5 j´jC ˇ jbj ŒnC1qŒ1C˛.ŒnC1q 1/…
Theorem 12. Let f .´/ 2 G ˛ n .;ˇ;b;q/: Then j´j ˇ jbj ŒnC1qŒ1C˛.ŒnC1q 1/ j´jnC1 5 ˇˇˇ˝ q;´f .´/ ˇˇˇ 5 j´jC ˇ jbj ŒnC1qŒ1C˛.ŒnC1q 1/ j´jnC1 : (5.19) The result is sharp for the function f .´/ given by D q;´f .´/ D q.´/´1
Corollary 14.
Corollary 14. Let f.´/ 2 G˛ n.;ˇ;b;q/: Then q.2/ q.2 / j´j1 1 ˇ jbj ŒnC1qŒ1C˛.ŒnC1q 1/ j´jn 5 ˇˇˇD q;´f.´/ ˇˇˇ
Corollary 14. Let f .´/ 2 G˛ n.;ˇ;b;q/: Then q.2/ q.2 / j´j1 1 ˇ jbj ŒnC1qŒ1C˛.ŒnC1q 1/ j´jn 5 ˇˇˇD q;´f .´/ ˇˇˇ
Corollary 15.
Corollary 15. Let f.´/ 2 G ˛ n.;ˇ;b;q/: Then q.2/ q.2C/ j´j1C 1 ˇ jbj ŒnC1qŒ1C˛.ŒnC1q 1/ j´jn 5 ˇˇˇI q;´f.´/ ˇˇˇ 5 q.2/…
Corollary 15. Let f .´/ 2 G ˛ n .;ˇ;b;q/: Then q.2/ q.2C/ j´j1C 1 ˇ jbj ŒnC1qŒ1C˛.ŒnC1q 1/ j´jn 5 ˇˇˇI q;´f .´/ ˇˇˇ 5 q.2/ q.2C/ j´j1C
Corollary 16.
Corollary 16. Let f.´/ 2 G ˛ n.;b;q/: Then q.2/ q.2 / j´j1 1 jbj ŒnC1qŒ1C˛.ŒnC1q 1/ j´jn 5 ˇˇˇD q;´f.´/ ˇˇˇ 5 q.2/ q.2 /…
Corollary 16. Let f .´/ 2 G ˛ n .;b;q/: Then q.2/ q.2 / j´j1 1 jbj ŒnC1qŒ1C˛.ŒnC1q 1/ j´jn 5 ˇˇˇD q;´f .´/ ˇˇˇ 5 q.2/ q.2 / j´j1
Corollary 17.
Corollary 17. Let f.´/ 2 G ˛ n.;ˇ;b;q/: Then q.2/ q.2C/ j´j1C 1 jbj ŒnC1qŒ1C˛.ŒnC1q 1/ j´jn 5 ˇˇˇI q;´f.´/ ˇˇˇ 5 q.2/…
Corollary 17. Let f .´/ 2 G ˛ n .;ˇ;b;q/: Then q.2/ q.2C/ j´j1C 1 jbj ŒnC1qŒ1C˛.ŒnC1q 1/ j´jn 5 ˇˇˇI q;´f .´/ ˇˇˇ 5 q.2/ q.2C/ j´j1C
Function classes studied:
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