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There is a 67 page review from last year, Log-concavity and strong log-concavity: a review, A. Saumard, J.A. Wellner (2014):

We review and formulate results concerning log-concavity and strong-log-concavity in both discrete and continuous settings. We show how preservation of log-concavity and strongly log-concavity on $\mathbb{R}$ under convolution follows from a fundamental monotonicity result of Efron (1969). We provide a new proof of Efron’s theorem using the recent asymmetric Brascamp-Lieb inequality due to Otto and Menz (2013). Along the way we review connections between log-concavity and other areas of mathematics and statistics, including concentration of measure, log-Sobolev inequalities, convex geometry, MCMC algorithms, Laplace approximations, and machine learning.

... and here are a whole bunch of older references:

1. A universal generator for discrete log-concave distributions, W Hörmann (1994).

2. A simple universal generator for continuous and discrete univariate T-concave distributions, J. Leydold (2001).

3. Preservation of log-concavity on summation, O. Johnson, C. Goldschmidt (2005).

4. Log-concavity and the maximum entropy property of the Poisson distribution, O. Johnson (2006).

5. On the entropy and log-concavity of compound Poisson measures, O. Johnson, I. Kontoyiannis, M. Madiman (2008).

6. Log-concavity, ultra-log-concavity, and a maximum entropy property of discrete compound Poisson measures, O. Johnson, I. Kontoyiannis, M. Madiman (2009).

7. Strong log-concavity is preserved by convolution, J.A. Wellner (2010).

8. Asymptotics of the discrete log‐concave maximum likelihood estimator and related applications, F. Balabdaoui, H. Jankowski, K. Rufibach, M. Pavlides, (2011).

There is a 67 page review from last year, Log-concavity and strong log-concavity: a review, A. Saumard, J.A. Wellner (2014):

We review and formulate results concerning log-concavity and strong-log-concavity in both discrete and continuous settings. We show how preservation of log-concavity and strongly log-concavity on $\mathbb{R}$ under convolution follows from a fundamental monotonicity result of Efron (1969). We provide a new proof of Efron’s theorem using the recent asymmetric Brascamp-Lieb inequality due to Otto and Menz (2013). Along the way we review connections between log-concavity and other areas of mathematics and statistics, including concentration of measure, log-Sobolev inequalities, convex geometry, MCMC algorithms, Laplace approximations, and machine learning.

This review contains many references, including some to older reviews and monographs. A few references are listed here, with hyperlinks:

1. A universal generator for discrete log-concave distributions, W Hörmann (1994).

2. A simple universal generator for continuous and discrete univariate T-concave distributions, J. Leydold (2001).

3. Preservation of log-concavity on summation, O. Johnson, C. Goldschmidt (2005).

4. Log-concavity and the maximum entropy property of the Poisson distribution, O. Johnson (2006).

5. On the entropy and log-concavity of compound Poisson measures, O. Johnson, I. Kontoyiannis, M. Madiman (2008).

6. Log-concavity, ultra-log-concavity, and a maximum entropy property of discrete compound Poisson measures, O. Johnson, I. Kontoyiannis, M. Madiman (2009).

7. Strong log-concavity is preserved by convolution, J.A. Wellner (2010).

8. Asymptotics of the discrete log‐concave maximum likelihood estimator and related applications, F. Balabdaoui, H. Jankowski, K. Rufibach, M. Pavlides, (2011).

1. A universal generator for discrete log-concave distributions, W Hörmann (1994).

2. A simple universal generator for continuous and discrete univariate T-concave distributions, J. Leydold (2001).

3. Preservation of log-concavity on summation, O. Johnson, C. Goldschmidt (2005).

4. Log-concavity and the maximum entropy property of the Poisson distribution, O. Johnson (2006).

5. On the entropy and log-concavity of compound Poisson measures, O. Johnson, I. Kontoyiannis, M. Madiman (2008).

6. Log-concavity, ultra-log-concavity, and a maximum entropy property of discrete compound Poisson measures, O. Johnson, I. Kontoyiannis, M. Madiman (2009).

7. Strong log-concavity is preserved by convolution, J.A. Wellner (2010).

8. Asymptotics of the discrete log‐concave maximum likelihood estimator and related applications, F. Balabdaoui, H. Jankowski, K. Rufibach, M. Pavlides, (2011).

There is a 67 page review from last year, Log-concavity and strong log-concavity: a review, A. Saumard, J.A. Wellner (2014):

We review and formulate results concerning log-concavity and strong-log-concavity in both discrete and continuous settings. We show how preservation of log-concavity and strongly log-concavity on $\mathbb{R}$ under convolution follows from a fundamental monotonicity result of Efron (1969). We provide a new proof of Efron’s theorem using the recent asymmetric Brascamp-Lieb inequality due to Otto and Menz (2013). Along the way we review connections between log-concavity and other areas of mathematics and statistics, including concentration of measure, log-Sobolev inequalities, convex geometry, MCMC algorithms, Laplace approximations, and machine learning.

... and here are a whole bunch of older references:

1. A universal generator for discrete log-concave distributions, W Hörmann (1994).

2. A simple universal generator for continuous and discrete univariate T-concave distributions, J. Leydold (2001).

3. Preservation of log-concavity on summation, O. Johnson, C. Goldschmidt (2005).

4. Log-concavity and the maximum entropy property of the Poisson distribution, O. Johnson (2006).

5. On the entropy and log-concavity of compound Poisson measures, O. Johnson, I. Kontoyiannis, M. Madiman (2008).

6. Log-concavity, ultra-log-concavity, and a maximum entropy property of discrete compound Poisson measures, O. Johnson, I. Kontoyiannis, M. Madiman (2009).

7. Strong log-concavity is preserved by convolution, J.A. Wellner (2010).

8. Asymptotics of the discrete log‐concave maximum likelihood estimator and related applications, F. Balabdaoui, H. Jankowski, K. Rufibach, M. Pavlides, (2011).

1. A universal generator for discrete log-concave distributions, W Hörmann (1994).

2. A simple universal generator for continuous and discrete univariate T-concave distributions, J. Leydold (2001).

3. Asymptotics of the discrete log‐concave maximum likelihood estimator and related applications, F. Balabdaoui, H. Jankowski, K. Rufibach, M. Pavlides, 2011.

4. Strong log-concavity is preserved by convolution, J.A. Wellner (2010).

5. Log-concavity and the maximum entropy property of the Poisson distribution, O. Johnson (2006).

6. On the entropy and log-concavity of compound Poisson measures, O. Johnson, I. Kontoyiannis, and M. Madiman (2008).

7. Log-concavity, ultra-log-concavity, and a maximum entropy property of discrete compound Poisson measures, O. Johnson, I. Kontoyiannis, and M. Madiman (2009).

1. A universal generator for discrete log-concave distributions, W Hörmann (1994).

2. A simple universal generator for continuous and discrete univariate T-concave distributions, J. Leydold (2001).

3. Preservation of log-concavity on summation, O. Johnson, C. Goldschmidt (2005).

4. Log-concavity and the maximum entropy property of the Poisson distribution, O. Johnson (2006).

5. On the entropy and log-concavity of compound Poisson measures, O. Johnson, I. Kontoyiannis, M. Madiman (2008).

6. Log-concavity, ultra-log-concavity, and a maximum entropy property of discrete compound Poisson measures, O. Johnson, I. Kontoyiannis, M. Madiman (2009).

7. Strong log-concavity is preserved by convolution, J.A. Wellner (2010).

8. Asymptotics of the discrete log‐concave maximum likelihood estimator and related applications, F. Balabdaoui, H. Jankowski, K. Rufibach, M. Pavlides, (2011).