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Medical images and patient information are routinely transmitted to a remote radiologist to assist in diagnosis. It is critical in e-healthcare systems to ensure that data are accurately transmitted. Medical images of a person’s body can be used against them in many ways, including by transmitting them. Copyright and intellectual property laws prohibit the unauthorized use of medical images. Digital watermarking is used to prove the authenticity of the medical images before diagnosis. In this paper, we proposed a hybrid watermarking scheme using the Slantlet transform, randomized-singular value decomposition, and optimization techniques inspired by nature (Firefly algorithm). The watermark image is encrypted using the XOR encryption technique. Extensive testing reveals that our innovative approach outperforms the existing methods based on the NC, SSIM, and PSNR. The SSIM and NC values of watermarked image and extracted watermark are close to or equal to 1 at a scaling factor of 0.06, and the PSNR of the proposed scheme lies between 58 dB and 59 dB, which shows the better performance of the scheme.

In this paper, we show that the power function of sub E-function f n (x) is sub E-function. Furthermore, we establish some new integral inequalities of Hadamard type involving sub E-functions and concave E-functions.