By Richard Courant, Charles de Prima, John R. Knudsen

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**Example text**

2 we conclude that Re − 4κ u (z) = Re u p+1 (z) > 0 for all z ∈ D. c p This means that u p is close-to-convex with respect to the convex function ϕ : D → C, defined by ϕ (z) = −(cz)/(4κ ). 3 it follows that u p is univalent in D. S. T. Mocanu [154]. 2 may be extended for complex parameters, as we can see in the following result. 3. (A. Re u p (z) > 0 for all z ∈ D. Further if Re κ ≥ |c|/4 and c = 0, then u p is univalent in D. Proof. The proof of this theorem is similar to the proof of the previous theorem.

24) z → I p (z) = 2 pΓ (p + 1)z−pI p (z) 16 1 Introduction and Preliminary Results was studied for example by C. Giordano et al. H. Ismail [119] and E. Neuman [159]. The function λ p : C → C, defined by λ p (z) = u p (z2 ), for c = b = 1 becomes J p and for c √ = −1, b = 1 it reduces to I p . Note that when b = 2 and c = 1 then w p becomes 2 j p / π and in this case λ p reduces to the function 1 z → J p+ 1 (z) = 2 p+ 2 Γ 2 p+ 3 −( p+ 1 ) 2 J z p+ 21 (z). 2 √ Similarly, when b = 2 and c = −1 then w p reduces to 2i p / π , and λ p in this case becomes 1 3 −( p+ 1 ) 2 I z z → I p+ 1 (z) = 2 p+ 2 Γ p + p+ 21 (z).

23) was studied by R. Askey [28], E. Neuman [161] and V. 24) z → I p (z) = 2 pΓ (p + 1)z−pI p (z) 16 1 Introduction and Preliminary Results was studied for example by C. Giordano et al. H. Ismail [119] and E. Neuman [159]. The function λ p : C → C, defined by λ p (z) = u p (z2 ), for c = b = 1 becomes J p and for c √ = −1, b = 1 it reduces to I p . Note that when b = 2 and c = 1 then w p becomes 2 j p / π and in this case λ p reduces to the function 1 z → J p+ 1 (z) = 2 p+ 2 Γ 2 p+ 3 −( p+ 1 ) 2 J z p+ 21 (z).

### Advanced methods in applied mathematics; lecture course (1941) by Richard Courant, Charles de Prima, John R. Knudsen

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